id	sid	tid	token	lemma	pos
ejpam-3781	1	1	european	european	PROPN
ejpam-3781	1	2	journal	journal	PROPN
ejpam-3781	1	3	of	of	ADP
ejpam-3781	1	4	pure	pure	ADJ
ejpam-3781	1	5	and	and	CCONJ
ejpam-3781	1	6	applied	apply	VERB
ejpam-3781	1	7	mathematics	mathematic	NOUN
ejpam-3781	1	8	vol	vol	NOUN
ejpam-3781	1	9	.	.	PUNCT
ejpam-3781	2	1	14	14	NUM
ejpam-3781	2	2	,	,	PUNCT
ejpam-3781	2	3	no	no	INTJ
ejpam-3781	2	4	.	.	NOUN
ejpam-3781	2	5	1	1	NUM
ejpam-3781	2	6	,	,	PUNCT
ejpam-3781	2	7	2021	2021	NUM
ejpam-3781	2	8	,	,	PUNCT
ejpam-3781	2	9	126	126	NUM
ejpam-3781	2	10	-	-	SYM
ejpam-3781	2	11	134	134	NUM
ejpam-3781	2	12	issn	issn	PROPN
ejpam-3781	2	13	1307	1307	NUM
ejpam-3781	2	14	-	-	SYM
ejpam-3781	2	15	5543	5543	NUM
ejpam-3781	2	16	–	–	PUNCT
ejpam-3781	3	1	ejpam.com	ejpam.com	X
ejpam-3781	3	2	published	publish	VERB
ejpam-3781	3	3	by	by	ADP
ejpam-3781	3	4	new	new	PROPN
ejpam-3781	3	5	york	york	PROPN
ejpam-3781	3	6	business	business	PROPN
ejpam-3781	3	7	global	global	PROPN
ejpam-3781	3	8	near	near	ADP
ejpam-3781	3	9	ring	ring	NOUN
ejpam-3781	3	10	multiplications	multiplication	NOUN
ejpam-3781	3	11	on	on	ADP
ejpam-3781	3	12	a	a	DET
ejpam-3781	3	13	modified	modify	VERB
ejpam-3781	3	14	near	near	ADP
ejpam-3781	3	15	module	module	NOUN
ejpam-3781	3	16	over	over	ADP
ejpam-3781	3	17	a	a	DET
ejpam-3781	3	18	near	near	ADJ
ejpam-3781	3	19	ring	ring	NOUN
ejpam-3781	3	20	a.	a.	NOUN
ejpam-3781	3	21	v.	v.	ADP
ejpam-3781	3	22	ramakrishna1,∗	ramakrishna1,∗	NOUN
ejpam-3781	3	23	,	,	PUNCT
ejpam-3781	3	24	t.v.n	t.v.n	NOUN
ejpam-3781	3	25	.	.	PUNCT
ejpam-3781	4	1	prasanna2	prasanna2	PROPN
ejpam-3781	4	2	,	,	PUNCT
ejpam-3781	4	3	,	,	PUNCT
ejpam-3781	4	4	d.v	d.v	PROPN
ejpam-3781	4	5	.	.	PUNCT
ejpam-3781	4	6	lakshmi3	lakshmi3	PROPN
ejpam-3781	4	7	1	1	NUM
ejpam-3781	4	8	department	department	NOUN
ejpam-3781	4	9	of	of	ADP
ejpam-3781	4	10	mathematics	mathematic	NOUN
ejpam-3781	4	11	,	,	PUNCT
ejpam-3781	4	12	r.v.r	r.v.r	NOUN
ejpam-3781	4	13	and	and	CCONJ
ejpam-3781	4	14	j.c	j.c	PROPN
ejpam-3781	4	15	college	college	PROPN
ejpam-3781	4	16	of	of	ADP
ejpam-3781	4	17	engineering	engineering	PROPN
ejpam-3781	4	18	,	,	PUNCT
ejpam-3781	4	19	chowdavaram	chowdavaram	PROPN
ejpam-3781	4	20	,	,	PUNCT
ejpam-3781	4	21	guntur-522019,andhra	guntur-522019,andhra	PROPN
ejpam-3781	4	22	pradesh	pradesh	PROPN
ejpam-3781	4	23	,	,	PUNCT
ejpam-3781	4	24	india	india	PROPN
ejpam-3781	4	25	2	2	NUM
ejpam-3781	4	26	department	department	PROPN
ejpam-3781	4	27	of	of	ADP
ejpam-3781	4	28	bs&h	bs&h	PROPN
ejpam-3781	4	29	,	,	PUNCT
ejpam-3781	4	30	vignan	vignan	NOUN
ejpam-3781	4	31	’s	’s	PART
ejpam-3781	4	32	nirula	nirula	PROPN
ejpam-3781	4	33	institute	institute	PROPN
ejpam-3781	4	34	of	of	ADP
ejpam-3781	4	35	technology	technology	PROPN
ejpam-3781	4	36	&	&	CCONJ
ejpam-3781	4	37	science	science	NOUN
ejpam-3781	4	38	for	for	ADP
ejpam-3781	4	39	women	woman	NOUN
ejpam-3781	4	40	,	,	PUNCT
ejpam-3781	4	41	guntur-522005	guntur-522005	VERB
ejpam-3781	4	42	,	,	PUNCT
ejpam-3781	4	43	andhra	andhra	PROPN
ejpam-3781	4	44	pradesh	pradesh	PROPN
ejpam-3781	4	45	,	,	PUNCT
ejpam-3781	4	46	india	india	PROPN
ejpam-3781	4	47	3	3	NUM
ejpam-3781	4	48	bapatla	bapatla	VERB
ejpam-3781	4	49	women	woman	NOUN
ejpam-3781	4	50	’s	’s	PART
ejpam-3781	4	51	engineering	engineering	PROPN
ejpam-3781	4	52	college	college	NOUN
ejpam-3781	4	53	,	,	PUNCT
ejpam-3781	4	54	bapatla	bapatla	NOUN
ejpam-3781	4	55	,	,	PUNCT
ejpam-3781	4	56	andhra	andhra	PROPN
ejpam-3781	4	57	pradesh	pradesh	PROPN
ejpam-3781	4	58	,	,	PUNCT
ejpam-3781	4	59	india	india	PROPN
ejpam-3781	4	60	abstract	abstract	NOUN
ejpam-3781	4	61	.	.	PUNCT
ejpam-3781	5	1	we	we	PRON
ejpam-3781	5	2	introduce	introduce	VERB
ejpam-3781	5	3	the	the	DET
ejpam-3781	5	4	notion	notion	NOUN
ejpam-3781	5	5	of	of	ADP
ejpam-3781	5	6	a	a	DET
ejpam-3781	5	7	modified	modify	VERB
ejpam-3781	5	8	near	near	ADP
ejpam-3781	5	9	module	module	NOUN
ejpam-3781	5	10	m	m	NOUN
ejpam-3781	5	11	over	over	ADP
ejpam-3781	5	12	a	a	DET
ejpam-3781	5	13	near	near	ADJ
ejpam-3781	5	14	ring	ring	NOUN
ejpam-3781	5	15	n	n	NOUN
ejpam-3781	5	16	and	and	CCONJ
ejpam-3781	5	17	explain	explain	VERB
ejpam-3781	5	18	a	a	DET
ejpam-3781	5	19	method	method	NOUN
ejpam-3781	5	20	of	of	ADP
ejpam-3781	5	21	obtaining	obtain	VERB
ejpam-3781	5	22	near	near	ADP
ejpam-3781	5	23	ring	ring	NOUN
ejpam-3781	5	24	multiplications	multiplication	NOUN
ejpam-3781	5	25	via	via	ADP
ejpam-3781	5	26	a	a	DET
ejpam-3781	5	27	special	special	ADJ
ejpam-3781	5	28	type	type	NOUN
ejpam-3781	5	29	of	of	ADP
ejpam-3781	5	30	maps	map	NOUN
ejpam-3781	5	31	from	from	ADP
ejpam-3781	5	32	m	m	PROPN
ejpam-3781	5	33	into	into	ADP
ejpam-3781	5	34	n	n	PROPN
ejpam-3781	5	35	called	call	VERB
ejpam-3781	5	36	semilinear	semilinear	NOUN
ejpam-3781	5	37	maps	map	NOUN
ejpam-3781	5	38	.	.	PUNCT
ejpam-3781	6	1	2020	2020	NUM
ejpam-3781	6	2	mathematics	mathematic	NOUN
ejpam-3781	6	3	subject	subject	NOUN
ejpam-3781	6	4	classifications	classification	NOUN
ejpam-3781	6	5	:	:	PUNCT
ejpam-3781	6	6	16y30	16y30	NUM
ejpam-3781	6	7	key	key	ADJ
ejpam-3781	6	8	words	word	NOUN
ejpam-3781	6	9	and	and	CCONJ
ejpam-3781	6	10	phrases	phrase	NOUN
ejpam-3781	6	11	:	:	PUNCT
ejpam-3781	6	12	near	near	ADP
ejpam-3781	6	13	ring	ring	NOUN
ejpam-3781	6	14	,	,	PUNCT
ejpam-3781	6	15	near	near	ADP
ejpam-3781	6	16	module	module	NOUN
ejpam-3781	6	17	,	,	PUNCT
ejpam-3781	6	18	semilinear	semilinear	NOUN
ejpam-3781	6	19	map	map	NOUN
ejpam-3781	6	20	1	1	NUM
ejpam-3781	6	21	.	.	X
ejpam-3781	6	22	introduction	introduction	NOUN
ejpam-3781	6	23	an	an	DET
ejpam-3781	6	24	interesting	interesting	ADJ
ejpam-3781	6	25	question	question	NOUN
ejpam-3781	6	26	that	that	PRON
ejpam-3781	6	27	has	have	AUX
ejpam-3781	6	28	attracted	attract	VERB
ejpam-3781	6	29	the	the	DET
ejpam-3781	6	30	attention	attention	NOUN
ejpam-3781	6	31	of	of	ADP
ejpam-3781	6	32	a	a	DET
ejpam-3781	6	33	good	good	ADJ
ejpam-3781	6	34	number	number	NOUN
ejpam-3781	6	35	of	of	ADP
ejpam-3781	6	36	near	near	ADJ
ejpam-3781	6	37	ring	ring	NOUN
ejpam-3781	6	38	theorists	theorist	NOUN
ejpam-3781	6	39	includes	include	VERB
ejpam-3781	6	40	j.r.clay	j.r.clay	NOUN
ejpam-3781	6	41	,	,	PUNCT
ejpam-3781	6	42	r.e.williams	r.e.william	NOUN
ejpam-3781	6	43	,	,	PUNCT
ejpam-3781	6	44	c.j.maxson	c.j.maxson	PROPN
ejpam-3781	6	45	,	,	PUNCT
ejpam-3781	6	46	m.johnson	m.johnson	PROPN
ejpam-3781	6	47	,	,	PUNCT
ejpam-3781	6	48	k.d.magill	k.d.magill	NOUN
ejpam-3781	6	49	jr.concerns	jr.concern	NOUN
ejpam-3781	6	50	with	with	ADP
ejpam-3781	6	51	finding	find	VERB
ejpam-3781	6	52	a	a	DET
ejpam-3781	6	53	near	near	ADJ
ejpam-3781	6	54	ring	ring	NOUN
ejpam-3781	6	55	multiplication	multiplication	NOUN
ejpam-3781	6	56	on	on	ADP
ejpam-3781	6	57	an	an	DET
ejpam-3781	6	58	algebraic	algebraic	ADJ
ejpam-3781	6	59	structure	structure	NOUN
ejpam-3781	6	60	over	over	ADP
ejpam-3781	6	61	an	an	DET
ejpam-3781	6	62	underlying	underlie	VERB
ejpam-3781	6	63	group	group	NOUN
ejpam-3781	6	64	.	.	PUNCT
ejpam-3781	7	1	in	in	ADP
ejpam-3781	7	2	particular	particular	PROPN
ejpam-3781	7	3	j.r	j.r	PROPN
ejpam-3781	7	4	.	.	PROPN
ejpam-3781	7	5	clay	clay	NOUN
ejpam-3781	7	6	(	(	PUNCT
ejpam-3781	7	7	1992	1992	NUM
ejpam-3781	7	8	)	)	PUNCT
ejpam-3781	8	1	[	[	X
ejpam-3781	8	2	6	6	NUM
ejpam-3781	8	3	]	]	PUNCT
ejpam-3781	8	4	proved	prove	VERB
ejpam-3781	8	5	that	that	SCONJ
ejpam-3781	8	6	a	a	DET
ejpam-3781	8	7	function	function	NOUN
ejpam-3781	8	8	π	π	X
ejpam-3781	8	9	on	on	ADP
ejpam-3781	8	10	a	a	DET
ejpam-3781	8	11	finite	finite	ADJ
ejpam-3781	8	12	cyclic	cyclic	NOUN
ejpam-3781	8	13	group	group	NOUN
ejpam-3781	8	14	(	(	PUNCT
ejpam-3781	8	15	zn,+	zn,+	PROPN
ejpam-3781	8	16	)	)	PUNCT
ejpam-3781	8	17	generates	generate	VERB
ejpam-3781	8	18	a	a	DET
ejpam-3781	8	19	multiplication	multiplication	NOUN
ejpam-3781	8	20	‘	'	PUNCT
ejpam-3781	8	21	*	*	NOUN
ejpam-3781	8	22	’	'	PUNCT
ejpam-3781	8	23	so	so	SCONJ
ejpam-3781	8	24	that	that	SCONJ
ejpam-3781	8	25	(	(	PUNCT
ejpam-3781	8	26	zn,+	zn,+	NUM
ejpam-3781	8	27	,	,	PUNCT
ejpam-3781	8	28	∗	∗	NOUN
ejpam-3781	8	29	)	)	PUNCT
ejpam-3781	8	30	is	be	AUX
ejpam-3781	8	31	a	a	DET
ejpam-3781	8	32	near	near	ADJ
ejpam-3781	8	33	ring	ring	NOUN
ejpam-3781	8	34	if	if	SCONJ
ejpam-3781	8	35	π(π(p)q	π(π(p)q	ADJ
ejpam-3781	8	36	)	)	PUNCT
ejpam-3781	8	37	=	=	SYM
ejpam-3781	8	38	π(p)π(q	π(p)π(q	NUM
ejpam-3781	8	39	)	)	PUNCT
ejpam-3781	8	40	.	.	PUNCT
ejpam-3781	9	1	k.d	k.d	PROPN
ejpam-3781	9	2	.	.	PROPN
ejpam-3781	9	3	magill	magill	PROPN
ejpam-3781	9	4	,	,	PUNCT
ejpam-3781	9	5	jr	jr	PROPN
ejpam-3781	9	6	.	.	PROPN
ejpam-3781	9	7	(	(	PUNCT
ejpam-3781	9	8	1995	1995	NUM
ejpam-3781	9	9	)	)	PUNCT
ejpam-3781	10	1	[	[	X
ejpam-3781	10	2	4	4	X
ejpam-3781	10	3	]	]	PUNCT
ejpam-3781	10	4	characterized	characterize	VERB
ejpam-3781	10	5	that	that	SCONJ
ejpam-3781	10	6	any	any	DET
ejpam-3781	10	7	near	near	ADP
ejpam-3781	10	8	ring	ring	NOUN
ejpam-3781	10	9	multiplication	multiplication	NOUN
ejpam-3781	10	10	on	on	ADP
ejpam-3781	10	11	a	a	DET
ejpam-3781	10	12	real	real	ADJ
ejpam-3781	10	13	finite	finite	ADJ
ejpam-3781	10	14	dimensional	dimensional	ADJ
ejpam-3781	10	15	euclidean	euclidean	ADJ
ejpam-3781	10	16	space	space	NOUN
ejpam-3781	10	17	rn	rn	PROPN
ejpam-3781	10	18	is	be	AUX
ejpam-3781	10	19	associated	associate	VERB
ejpam-3781	10	20	with	with	ADP
ejpam-3781	10	21	a	a	DET
ejpam-3781	10	22	real	real	ADV
ejpam-3781	10	23	-	-	PUNCT
ejpam-3781	10	24	valued	value	VERB
ejpam-3781	10	25	function	function	NOUN
ejpam-3781	10	26	f	f	PROPN
ejpam-3781	10	27	on	on	ADP
ejpam-3781	10	28	rn	rn	PROPN
ejpam-3781	10	29	that	that	SCONJ
ejpam-3781	10	30	satisfies	satisfy	VERB
ejpam-3781	10	31	f(f(x)y	f(f(x)y	NUM
ejpam-3781	10	32	)	)	PUNCT
ejpam-3781	10	33	=	=	SYM
ejpam-3781	10	34	f(x)f(y	f(x)f(y	NOUN
ejpam-3781	10	35	)	)	PUNCT
ejpam-3781	10	36	.	.	PUNCT
ejpam-3781	11	1	in	in	ADP
ejpam-3781	11	2	this	this	DET
ejpam-3781	11	3	paper	paper	NOUN
ejpam-3781	11	4	we	we	PRON
ejpam-3781	11	5	present	present	VERB
ejpam-3781	11	6	methods	method	NOUN
ejpam-3781	11	7	[	[	X
ejpam-3781	11	8	2	2	NUM
ejpam-3781	11	9	]	]	PUNCT
ejpam-3781	11	10	,	,	PUNCT
ejpam-3781	11	11	[	[	X
ejpam-3781	11	12	3	3	X
ejpam-3781	11	13	]	]	X
ejpam-3781	12	1	[	[	X
ejpam-3781	12	2	1	1	NUM
ejpam-3781	12	3	]	]	PUNCT
ejpam-3781	12	4	of	of	ADP
ejpam-3781	12	5	finding	find	VERB
ejpam-3781	12	6	near	near	ADP
ejpam-3781	12	7	ring	ring	NOUN
ejpam-3781	12	8	multiplications	multiplication	NOUN
ejpam-3781	12	9	on	on	ADP
ejpam-3781	12	10	some	some	DET
ejpam-3781	12	11	algebraic	algebraic	ADJ
ejpam-3781	12	12	structures	structure	NOUN
ejpam-3781	12	13	which	which	PRON
ejpam-3781	12	14	we	we	PRON
ejpam-3781	12	15	call	call	VERB
ejpam-3781	12	16	modified	modify	VERB
ejpam-3781	12	17	near	near	ADP
ejpam-3781	12	18	modules	module	NOUN
ejpam-3781	12	19	.	.	PUNCT
ejpam-3781	13	1	a	a	DET
ejpam-3781	13	2	right	right	NOUN
ejpam-3781	13	3	near	near	ADP
ejpam-3781	13	4	ring	ring	NOUN
ejpam-3781	13	5	[	[	X
ejpam-3781	13	6	5]is	5]is	NUM
ejpam-3781	13	7	a	a	DET
ejpam-3781	13	8	triple	triple	ADJ
ejpam-3781	13	9	(	(	PUNCT
ejpam-3781	13	10	n,+	n,+	NUM
ejpam-3781	13	11	,	,	PUNCT
ejpam-3781	13	12	.	.	PUNCT
ejpam-3781	13	13	)	)	PUNCT
ejpam-3781	13	14	,	,	PUNCT
ejpam-3781	13	15	where	where	SCONJ
ejpam-3781	13	16	(	(	PUNCT
ejpam-3781	13	17	n,+	n,+	NUM
ejpam-3781	13	18	)	)	PUNCT
ejpam-3781	13	19	is	be	AUX
ejpam-3781	13	20	a	a	DET
ejpam-3781	13	21	(	(	PUNCT
ejpam-3781	13	22	not	not	PART
ejpam-3781	13	23	necessarily	necessarily	ADV
ejpam-3781	13	24	abelian	abelian	ADJ
ejpam-3781	13	25	)	)	PUNCT
ejpam-3781	13	26	group	group	NOUN
ejpam-3781	13	27	,	,	PUNCT
ejpam-3781	13	28	(	(	PUNCT
ejpam-3781	13	29	n	n	CCONJ
ejpam-3781	13	30	,	,	PUNCT
ejpam-3781	13	31	.	.	PUNCT
ejpam-3781	13	32	)	)	PUNCT
ejpam-3781	14	1	is	be	AUX
ejpam-3781	14	2	a	a	DET
ejpam-3781	14	3	semigroup	semigroup	NOUN
ejpam-3781	14	4	satisfying	satisfy	VERB
ejpam-3781	14	5	the	the	DET
ejpam-3781	14	6	right	right	ADJ
ejpam-3781	14	7	distributive	distributive	ADJ
ejpam-3781	14	8	law	law	NOUN
ejpam-3781	14	9	:	:	PUNCT
ejpam-3781	14	10	(	(	PUNCT
ejpam-3781	14	11	a+	a+	X
ejpam-3781	14	12	b)c	b)c	X
ejpam-3781	14	13	=	=	NUM
ejpam-3781	14	14	ac+	ac+	PROPN
ejpam-3781	14	15	bc	bc	PROPN
ejpam-3781	14	16	for	for	ADP
ejpam-3781	14	17	all	all	DET
ejpam-3781	14	18	a	a	DET
ejpam-3781	14	19	,	,	PUNCT
ejpam-3781	14	20	b	b	NOUN
ejpam-3781	14	21	,	,	PUNCT
ejpam-3781	14	22	c	c	PROPN
ejpam-3781	14	23	∈	∈	PROPN
ejpam-3781	14	24	n	n	ADV
ejpam-3781	14	25	.	.	PUNCT
ejpam-3781	15	1	by	by	ADP
ejpam-3781	15	2	a	a	DET
ejpam-3781	15	3	near	near	ADJ
ejpam-3781	15	4	ring	ring	NOUN
ejpam-3781	15	5	we	we	PRON
ejpam-3781	15	6	mean	mean	VERB
ejpam-3781	15	7	a	a	DET
ejpam-3781	15	8	right	right	NOUN
ejpam-3781	15	9	near	near	ADP
ejpam-3781	15	10	ring	ring	NOUN
ejpam-3781	15	11	.	.	PUNCT
ejpam-3781	16	1	when	when	SCONJ
ejpam-3781	16	2	there	there	PRON
ejpam-3781	16	3	is	be	VERB
ejpam-3781	16	4	no	no	DET
ejpam-3781	16	5	scope	scope	NOUN
ejpam-3781	16	6	for	for	ADP
ejpam-3781	16	7	confusion	confusion	NOUN
ejpam-3781	16	8	,	,	PUNCT
ejpam-3781	16	9	we	we	PRON
ejpam-3781	16	10	write	write	VERB
ejpam-3781	16	11	n	n	PRON
ejpam-3781	16	12	is	be	AUX
ejpam-3781	16	13	a	a	DET
ejpam-3781	16	14	near	near	ADJ
ejpam-3781	16	15	ring	ring	NOUN
ejpam-3781	16	16	instead	instead	ADV
ejpam-3781	16	17	of	of	ADP
ejpam-3781	16	18	(	(	PUNCT
ejpam-3781	16	19	n,+	n,+	NUM
ejpam-3781	16	20	,	,	PUNCT
ejpam-3781	16	21	·	·	PUNCT
ejpam-3781	16	22	)	)	PUNCT
ejpam-3781	16	23	is	be	AUX
ejpam-3781	16	24	a	a	DET
ejpam-3781	16	25	near	near	ADJ
ejpam-3781	16	26	ring	ring	NOUN
ejpam-3781	16	27	.	.	PUNCT
ejpam-3781	17	1	[	[	X
ejpam-3781	17	2	6	6	NUM
ejpam-3781	17	3	]	]	PUNCT
ejpam-3781	17	4	∗corresponding	∗corresponde	VERB
ejpam-3781	17	5	author	author	NOUN
ejpam-3781	17	6	.	.	PUNCT
ejpam-3781	18	1	doi	doi	NOUN
ejpam-3781	18	2	:	:	PUNCT
ejpam-3781	18	3	https://doi.org/10.29020/nybg.ejpam.v14i1.3781	https://doi.org/10.29020/nybg.ejpam.v14i1.3781	NUM
ejpam-3781	18	4	email	email	NOUN
ejpam-3781	18	5	addresses	address	NOUN
ejpam-3781	18	6	:	:	PUNCT
ejpam-3781	18	7	amathi7@gmail.com	amathi7@gmail.com	X
ejpam-3781	18	8	(	(	PUNCT
ejpam-3781	18	9	a.	a.	PROPN
ejpam-3781	18	10	v.	v.	PROPN
ejpam-3781	18	11	ramakrishna	ramakrishna	PROPN
ejpam-3781	18	12	)	)	PUNCT
ejpam-3781	18	13	,	,	PUNCT
ejpam-3781	18	14	tvnp11@gmail.com	tvnp11@gmail.com	X
ejpam-3781	18	15	(	(	PUNCT
ejpam-3781	18	16	t.v.n	t.v.n	PROPN
ejpam-3781	18	17	.	.	PUNCT
ejpam-3781	18	18	prasanna	prasanna	PROPN
ejpam-3781	18	19	)	)	PUNCT
ejpam-3781	18	20	,	,	PUNCT
ejpam-3781	18	21	himaja96@gmail.com	himaja96@gmail.com	X
ejpam-3781	18	22	(	(	PUNCT
ejpam-3781	18	23	d.v	d.v	PROPN
ejpam-3781	18	24	.	.	PROPN
ejpam-3781	18	25	lakshmi	lakshmi	PROPN
ejpam-3781	18	26	)	)	PUNCT
ejpam-3781	18	27	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3781	19	1	126	126	NUM
ejpam-3781	19	2	c	c	X
ejpam-3781	19	3	©	©	NOUN
ejpam-3781	19	4	2021	2021	NUM
ejpam-3781	19	5	ejpam	ejpam	VERB
ejpam-3781	19	6	all	all	DET
ejpam-3781	19	7	rights	right	NOUN
ejpam-3781	19	8	reserved	reserve	VERB
ejpam-3781	19	9	.	.	PUNCT
ejpam-3781	20	1	a.v	a.v	PROPN
ejpam-3781	20	2	.	.	PROPN
ejpam-3781	20	3	ramakrishna	ramakrishna	PROPN
ejpam-3781	20	4	,	,	PUNCT
ejpam-3781	20	5	t.v.n	t.v.n	PROPN
ejpam-3781	20	6	.	.	PUNCT
ejpam-3781	21	1	prasanna	prasanna	PROPN
ejpam-3781	21	2	,	,	PUNCT
ejpam-3781	21	3	d.v	d.v	PROPN
ejpam-3781	21	4	.	.	PROPN
ejpam-3781	21	5	lakshmi	lakshmi	PROPN
ejpam-3781	21	6	/	/	SYM
ejpam-3781	21	7	eur	eur	PROPN
ejpam-3781	21	8	.	.	PUNCT
ejpam-3781	22	1	j.	j.	PROPN
ejpam-3781	22	2	pure	pure	PROPN
ejpam-3781	22	3	appl	appl	PROPN
ejpam-3781	22	4	.	.	PROPN
ejpam-3781	22	5	math	math	PROPN
ejpam-3781	22	6	,	,	PUNCT
ejpam-3781	22	7	14	14	NUM
ejpam-3781	22	8	(	(	PUNCT
ejpam-3781	22	9	1	1	NUM
ejpam-3781	22	10	)	)	PUNCT
ejpam-3781	22	11	(	(	PUNCT
ejpam-3781	22	12	2021	2021	NUM
ejpam-3781	22	13	)	)	PUNCT
ejpam-3781	22	14	,	,	PUNCT
ejpam-3781	22	15	126	126	NUM
ejpam-3781	22	16	-	-	SYM
ejpam-3781	22	17	134	134	NUM
ejpam-3781	22	18	127	127	NUM
ejpam-3781	22	19	2	2	NUM
ejpam-3781	22	20	.	.	PUNCT
ejpam-3781	22	21	modified	modify	VERB
ejpam-3781	22	22	near	near	ADP
ejpam-3781	22	23	modules	module	NOUN
ejpam-3781	22	24	let	let	VERB
ejpam-3781	22	25	(	(	PUNCT
ejpam-3781	22	26	m,+	m,+	NUM
ejpam-3781	22	27	)	)	PUNCT
ejpam-3781	22	28	be	be	AUX
ejpam-3781	22	29	a	a	DET
ejpam-3781	22	30	group	group	NOUN
ejpam-3781	22	31	and	and	CCONJ
ejpam-3781	22	32	let	let	VERB
ejpam-3781	22	33	n	n	PRON
ejpam-3781	22	34	be	be	AUX
ejpam-3781	22	35	a	a	DET
ejpam-3781	22	36	near	near	ADJ
ejpam-3781	22	37	ring	ring	NOUN
ejpam-3781	22	38	and	and	CCONJ
ejpam-3781	22	39	suppose	suppose	VERB
ejpam-3781	22	40	‘	'	PUNCT
ejpam-3781	22	41	·	·	PUNCT
ejpam-3781	22	42	’	'	PUNCT
ejpam-3781	22	43	is	be	AUX
ejpam-3781	22	44	a	a	DET
ejpam-3781	22	45	mapping	mapping	NOUN
ejpam-3781	22	46	of	of	ADP
ejpam-3781	22	47	n×m	n×m	PROPN
ejpam-3781	22	48	into	into	ADP
ejpam-3781	22	49	m	m	PROPN
ejpam-3781	22	50	.	.	PUNCT
ejpam-3781	23	1	definition	definition	NOUN
ejpam-3781	23	2	1	1	NUM
ejpam-3781	23	3	.	.	PUNCT
ejpam-3781	24	1	(	(	PUNCT
ejpam-3781	24	2	m,+	m,+	PROPN
ejpam-3781	24	3	,	,	PUNCT
ejpam-3781	24	4	·	·	PUNCT
ejpam-3781	24	5	)	)	PUNCT
ejpam-3781	24	6	is	be	AUX
ejpam-3781	24	7	called	call	VERB
ejpam-3781	24	8	a	a	DET
ejpam-3781	24	9	near	near	ADJ
ejpam-3781	24	10	module	module	NOUN
ejpam-3781	24	11	over	over	ADP
ejpam-3781	24	12	n	n	NOUN
ejpam-3781	24	13	if	if	SCONJ
ejpam-3781	24	14	(	(	PUNCT
ejpam-3781	24	15	i	i	NOUN
ejpam-3781	24	16	)	)	PUNCT
ejpam-3781	24	17	(	(	PUNCT
ejpam-3781	24	18	n1	n1	NOUN
ejpam-3781	24	19	+	+	CCONJ
ejpam-3781	24	20	n2)m	n2)m	NOUN
ejpam-3781	24	21	=	=	SYM
ejpam-3781	24	22	n1m+	n1m+	PROPN
ejpam-3781	24	23	n2	n2	NOUN
ejpam-3781	24	24	m	m	VERB
ejpam-3781	24	25	for	for	ADP
ejpam-3781	24	26	all	all	DET
ejpam-3781	24	27	n1	n1	NOUN
ejpam-3781	24	28	,	,	PUNCT
ejpam-3781	24	29	n2	n2	NOUN
ejpam-3781	24	30	∈	∈	PROPN
ejpam-3781	24	31	n	n	PRON
ejpam-3781	24	32	and	and	CCONJ
ejpam-3781	24	33	m	m	NOUN
ejpam-3781	24	34	∈m	∈m	NOUN
ejpam-3781	24	35	;	;	PUNCT
ejpam-3781	24	36	(	(	PUNCT
ejpam-3781	24	37	ii	ii	NOUN
ejpam-3781	24	38	)	)	PUNCT
ejpam-3781	24	39	(	(	PUNCT
ejpam-3781	24	40	n1n2)m	n1n2)m	X
ejpam-3781	24	41	=	=	PUNCT
ejpam-3781	24	42	n1(n2	n1(n2	NOUN
ejpam-3781	24	43	m	m	VERB
ejpam-3781	24	44	)	)	PUNCT
ejpam-3781	24	45	for	for	ADP
ejpam-3781	24	46	all	all	DET
ejpam-3781	24	47	n1	n1	NOUN
ejpam-3781	24	48	,	,	PUNCT
ejpam-3781	24	49	n2	n2	NOUN
ejpam-3781	24	50	∈	∈	PROPN
ejpam-3781	24	51	n	n	PRON
ejpam-3781	24	52	and	and	CCONJ
ejpam-3781	24	53	m	m	PRON
ejpam-3781	24	54	∈m	∈m	NOUN
ejpam-3781	24	55	.	.	PUNCT
ejpam-3781	25	1	remark	remark	PROPN
ejpam-3781	25	2	1	1	NUM
ejpam-3781	25	3	.	.	PUNCT
ejpam-3781	26	1	clearly	clearly	ADV
ejpam-3781	26	2	our	our	PRON
ejpam-3781	26	3	near	near	ADJ
ejpam-3781	26	4	module	module	NOUN
ejpam-3781	26	5	is	be	AUX
ejpam-3781	26	6	the	the	DET
ejpam-3781	26	7	n	n	NOUN
ejpam-3781	26	8	-	-	PUNCT
ejpam-3781	26	9	group	group	NOUN
ejpam-3781	26	10	introduced	introduce	VERB
ejpam-3781	26	11	by	by	ADP
ejpam-3781	26	12	pilz	pilz	PROPN
ejpam-3781	26	13	.	.	PUNCT
ejpam-3781	27	1	definition	definition	NOUN
ejpam-3781	27	2	2	2	NUM
ejpam-3781	27	3	.	.	PUNCT
ejpam-3781	27	4	(	(	PUNCT
ejpam-3781	27	5	m,+	m,+	PROPN
ejpam-3781	27	6	,	,	PUNCT
ejpam-3781	27	7	·	·	PUNCT
ejpam-3781	27	8	)	)	PUNCT
ejpam-3781	27	9	is	be	AUX
ejpam-3781	27	10	called	call	VERB
ejpam-3781	27	11	a	a	DET
ejpam-3781	27	12	modified	modify	VERB
ejpam-3781	27	13	near	near	ADP
ejpam-3781	27	14	module	module	NOUN
ejpam-3781	27	15	over	over	ADP
ejpam-3781	27	16	n	n	NOUN
ejpam-3781	27	17	if	if	SCONJ
ejpam-3781	27	18	(	(	PUNCT
ejpam-3781	27	19	i	i	NOUN
ejpam-3781	27	20	)	)	PUNCT
ejpam-3781	27	21	n(m1	n(m1	NOUN
ejpam-3781	28	1	+	+	NOUN
ejpam-3781	28	2	m2	m2	X
ejpam-3781	28	3	)	)	PUNCT
ejpam-3781	28	4	=	=	PUNCT
ejpam-3781	29	1	nm1	nm1	VERB
ejpam-3781	30	1	+	+	CCONJ
ejpam-3781	30	2	nm2	nm2	NOUN
ejpam-3781	30	3	for	for	ADP
ejpam-3781	30	4	all	all	PRON
ejpam-3781	30	5	n	n	PRON
ejpam-3781	30	6	∈	∈	PROPN
ejpam-3781	30	7	n	n	NOUN
ejpam-3781	30	8	and	and	CCONJ
ejpam-3781	30	9	m1,m2	m1,m2	PROPN
ejpam-3781	30	10	∈m	∈m	NOUN
ejpam-3781	30	11	;	;	PUNCT
ejpam-3781	30	12	(	(	PUNCT
ejpam-3781	30	13	ii	ii	NOUN
ejpam-3781	30	14	)	)	PUNCT
ejpam-3781	30	15	(	(	PUNCT
ejpam-3781	30	16	n1n2)m	n1n2)m	X
ejpam-3781	30	17	=	=	PUNCT
ejpam-3781	30	18	n1(n2	n1(n2	NOUN
ejpam-3781	30	19	m	m	VERB
ejpam-3781	30	20	)	)	PUNCT
ejpam-3781	30	21	for	for	ADP
ejpam-3781	30	22	all	all	DET
ejpam-3781	30	23	n1	n1	NOUN
ejpam-3781	30	24	,	,	PUNCT
ejpam-3781	30	25	n2	n2	NOUN
ejpam-3781	30	26	∈	∈	PROPN
ejpam-3781	30	27	n	n	PRON
ejpam-3781	30	28	and	and	CCONJ
ejpam-3781	30	29	m	m	PRON
ejpam-3781	30	30	∈m	∈m	NOUN
ejpam-3781	30	31	.	.	PUNCT
ejpam-3781	31	1	definition	definition	NOUN
ejpam-3781	31	2	3	3	NUM
ejpam-3781	31	3	.	.	PUNCT
ejpam-3781	32	1	(	(	PUNCT
ejpam-3781	32	2	m,+	m,+	PROPN
ejpam-3781	32	3	,	,	PUNCT
ejpam-3781	32	4	·	·	PUNCT
ejpam-3781	32	5	)	)	PUNCT
ejpam-3781	32	6	is	be	AUX
ejpam-3781	32	7	called	call	VERB
ejpam-3781	32	8	a	a	DET
ejpam-3781	32	9	strong	strong	ADJ
ejpam-3781	32	10	near	near	ADP
ejpam-3781	32	11	module	module	NOUN
ejpam-3781	32	12	over	over	ADP
ejpam-3781	32	13	n	n	NOUN
ejpam-3781	32	14	if	if	SCONJ
ejpam-3781	32	15	(	(	PUNCT
ejpam-3781	32	16	i	i	NOUN
ejpam-3781	32	17	)	)	PUNCT
ejpam-3781	32	18	(	(	PUNCT
ejpam-3781	32	19	n1	n1	NOUN
ejpam-3781	32	20	+	+	CCONJ
ejpam-3781	32	21	n2)m	n2)m	NOUN
ejpam-3781	32	22	=	=	SYM
ejpam-3781	32	23	n1m+	n1m+	PROPN
ejpam-3781	32	24	n2	n2	NOUN
ejpam-3781	32	25	m	m	VERB
ejpam-3781	32	26	for	for	ADP
ejpam-3781	32	27	all	all	DET
ejpam-3781	32	28	n1	n1	NOUN
ejpam-3781	32	29	,	,	PUNCT
ejpam-3781	32	30	n2	n2	NOUN
ejpam-3781	32	31	∈	∈	PROPN
ejpam-3781	32	32	n	n	PRON
ejpam-3781	32	33	and	and	CCONJ
ejpam-3781	32	34	m	m	NOUN
ejpam-3781	32	35	∈m	∈m	NOUN
ejpam-3781	32	36	;	;	PUNCT
ejpam-3781	32	37	(	(	PUNCT
ejpam-3781	32	38	ii	ii	NOUN
ejpam-3781	32	39	)	)	PUNCT
ejpam-3781	32	40	n(m1	n(m1	NOUN
ejpam-3781	33	1	+	+	NOUN
ejpam-3781	33	2	m2	m2	X
ejpam-3781	33	3	)	)	PUNCT
ejpam-3781	33	4	=	=	PUNCT
ejpam-3781	34	1	nm1	nm1	VERB
ejpam-3781	35	1	+	+	CCONJ
ejpam-3781	35	2	nm2	nm2	ADJ
ejpam-3781	35	3	,	,	PUNCT
ejpam-3781	35	4	for	for	ADP
ejpam-3781	35	5	all	all	DET
ejpam-3781	35	6	m1,m2	m1,m2	PROPN
ejpam-3781	35	7	∈m	∈m	NOUN
ejpam-3781	35	8	and	and	CCONJ
ejpam-3781	35	9	n	n	PRON
ejpam-3781	35	10	∈	∈	PROPN
ejpam-3781	35	11	n	n	NOUN
ejpam-3781	35	12	;	;	PUNCT
ejpam-3781	35	13	(	(	PUNCT
ejpam-3781	35	14	iii	iii	X
ejpam-3781	35	15	)	)	PUNCT
ejpam-3781	35	16	(	(	PUNCT
ejpam-3781	35	17	n1n2)m	n1n2)m	X
ejpam-3781	35	18	=	=	PUNCT
ejpam-3781	35	19	n1(n2	n1(n2	NOUN
ejpam-3781	35	20	m	m	VERB
ejpam-3781	35	21	)	)	PUNCT
ejpam-3781	35	22	for	for	ADP
ejpam-3781	35	23	all	all	DET
ejpam-3781	35	24	n1	n1	NOUN
ejpam-3781	35	25	,	,	PUNCT
ejpam-3781	35	26	n2	n2	NOUN
ejpam-3781	35	27	∈	∈	PROPN
ejpam-3781	35	28	n	n	PRON
ejpam-3781	35	29	and	and	CCONJ
ejpam-3781	35	30	m	m	PRON
ejpam-3781	35	31	∈m	∈m	NOUN
ejpam-3781	35	32	.	.	PUNCT
ejpam-3781	36	1	remark	remark	NOUN
ejpam-3781	36	2	2	2	NUM
ejpam-3781	36	3	.	.	PUNCT
ejpam-3781	37	1	a	a	DET
ejpam-3781	37	2	strong	strong	ADJ
ejpam-3781	37	3	near	near	ADP
ejpam-3781	37	4	module	module	NOUN
ejpam-3781	37	5	over	over	ADP
ejpam-3781	37	6	a	a	DET
ejpam-3781	37	7	field	field	NOUN
ejpam-3781	37	8	is	be	AUX
ejpam-3781	37	9	a	a	DET
ejpam-3781	37	10	vector	vector	NOUN
ejpam-3781	37	11	space	space	NOUN
ejpam-3781	37	12	if	if	SCONJ
ejpam-3781	37	13	‘	'	PUNCT
ejpam-3781	37	14	+	+	NOUN
ejpam-3781	37	15	’	'	PUNCT
ejpam-3781	37	16	is	be	AUX
ejpam-3781	37	17	abelian	abelian	ADJ
ejpam-3781	37	18	and	and	CCONJ
ejpam-3781	37	19	1	1	NUM
ejpam-3781	37	20	m	m	NOUN
ejpam-3781	37	21	=	=	VERB
ejpam-3781	37	22	m	m	VERB
ejpam-3781	37	23	for	for	ADP
ejpam-3781	37	24	every	every	DET
ejpam-3781	37	25	m.	m.	NOUN
ejpam-3781	37	26	example	example	NOUN
ejpam-3781	38	1	1	1	X
ejpam-3781	38	2	.	.	PUNCT
ejpam-3781	39	1	let	let	AUX
ejpam-3781	39	2	(	(	PUNCT
ejpam-3781	39	3	g,+	g,+	PROPN
ejpam-3781	39	4	)	)	PUNCT
ejpam-3781	39	5	be	be	AUX
ejpam-3781	39	6	a	a	DET
ejpam-3781	39	7	group	group	NOUN
ejpam-3781	39	8	.	.	PUNCT
ejpam-3781	40	1	define	define	VERB
ejpam-3781	40	2	the	the	DET
ejpam-3781	40	3	function	function	NOUN
ejpam-3781	40	4	·	·	PUNCT
ejpam-3781	40	5	from	from	ADP
ejpam-3781	40	6	m(g	m(g	PROPN
ejpam-3781	40	7	)	)	PUNCT
ejpam-3781	40	8	×	×	NOUN
ejpam-3781	40	9	g	g	NOUN
ejpam-3781	40	10	into	into	ADP
ejpam-3781	40	11	g	g	NOUN
ejpam-3781	40	12	by	by	ADP
ejpam-3781	40	13	·	·	PUNCT
ejpam-3781	40	14	(	(	PUNCT
ejpam-3781	40	15	f	f	NUM
ejpam-3781	40	16	,	,	PUNCT
ejpam-3781	40	17	x	x	NOUN
ejpam-3781	40	18	)	)	PUNCT
ejpam-3781	40	19	=	=	SYM
ejpam-3781	40	20	f	f	X
ejpam-3781	40	21	·	·	PUNCT
ejpam-3781	40	22	x	x	X
ejpam-3781	40	23	=	=	SYM
ejpam-3781	40	24	f(x	f(x	PROPN
ejpam-3781	40	25	)	)	PUNCT
ejpam-3781	40	26	for	for	ADP
ejpam-3781	40	27	all	all	DET
ejpam-3781	40	28	f	f	PROPN
ejpam-3781	40	29	∈m(g	∈m(g	PROPN
ejpam-3781	40	30	)	)	PUNCT
ejpam-3781	40	31	and	and	CCONJ
ejpam-3781	40	32	x	x	PUNCT
ejpam-3781	40	33	∈	∈	PROPN
ejpam-3781	40	34	g.	g.	NOUN
ejpam-3781	40	35	for	for	ADP
ejpam-3781	40	36	any	any	DET
ejpam-3781	40	37	f	f	NOUN
ejpam-3781	40	38	,	,	PUNCT
ejpam-3781	40	39	g	g	PROPN
ejpam-3781	40	40	∈m(g	∈m(g	PROPN
ejpam-3781	40	41	)	)	PUNCT
ejpam-3781	40	42	and	and	CCONJ
ejpam-3781	40	43	x	x	PUNCT
ejpam-3781	40	44	∈	∈	PROPN
ejpam-3781	40	45	g	g	PROPN
ejpam-3781	40	46	,	,	PUNCT
ejpam-3781	40	47	(	(	PUNCT
ejpam-3781	40	48	f	f	X
ejpam-3781	40	49	◦	◦	PROPN
ejpam-3781	40	50	g)(x	g)(x	PROPN
ejpam-3781	40	51	)	)	PUNCT
ejpam-3781	40	52	=	=	SYM
ejpam-3781	40	53	f(g(x	f(g(x	NOUN
ejpam-3781	40	54	)	)	PUNCT
ejpam-3781	40	55	)	)	PUNCT
ejpam-3781	41	1	=	=	SYM
ejpam-3781	41	2	f(gx	f(gx	NOUN
ejpam-3781	41	3	)	)	PUNCT
ejpam-3781	41	4	.	.	PUNCT
ejpam-3781	42	1	also	also	ADV
ejpam-3781	42	2	(	(	PUNCT
ejpam-3781	42	3	f	f	X
ejpam-3781	42	4	+	+	CCONJ
ejpam-3781	42	5	g)(x	g)(x	PROPN
ejpam-3781	42	6	)	)	PUNCT
ejpam-3781	42	7	=	=	SYM
ejpam-3781	42	8	f(x	f(x	PROPN
ejpam-3781	42	9	)	)	PUNCT
ejpam-3781	43	1	+	+	CCONJ
ejpam-3781	43	2	g(x	g(x	NOUN
ejpam-3781	43	3	)	)	PUNCT
ejpam-3781	43	4	=	=	SYM
ejpam-3781	43	5	fx+	fx+	PROPN
ejpam-3781	43	6	gx	gx	PROPN
ejpam-3781	43	7	.	.	PROPN
ejpam-3781	43	8	and	and	CCONJ
ejpam-3781	43	9	f(x+y	f(x+y	NUM
ejpam-3781	43	10	)	)	PUNCT
ejpam-3781	43	11	6=	6=	NUM
ejpam-3781	43	12	f(x)+f(y	f(x)+f(y	NOUN
ejpam-3781	43	13	)	)	PUNCT
ejpam-3781	43	14	.	.	PUNCT
ejpam-3781	44	1	therefore	therefore	ADV
ejpam-3781	44	2	(	(	PUNCT
ejpam-3781	44	3	g,+	g,+	PROPN
ejpam-3781	44	4	,	,	PUNCT
ejpam-3781	44	5	·	·	PUNCT
ejpam-3781	44	6	)	)	PUNCT
ejpam-3781	44	7	is	be	AUX
ejpam-3781	44	8	a	a	DET
ejpam-3781	44	9	near	near	ADJ
ejpam-3781	44	10	module	module	NOUN
ejpam-3781	44	11	over	over	ADP
ejpam-3781	44	12	near	near	ADP
ejpam-3781	44	13	ring	ring	NOUN
ejpam-3781	44	14	(	(	PUNCT
ejpam-3781	44	15	m(g),+	m(g),+	NOUN
ejpam-3781	44	16	,	,	PUNCT
ejpam-3781	44	17	◦	◦	NOUN
ejpam-3781	44	18	)	)	PUNCT
ejpam-3781	44	19	,	,	PUNCT
ejpam-3781	44	20	but	but	CCONJ
ejpam-3781	44	21	not	not	PART
ejpam-3781	44	22	a	a	DET
ejpam-3781	44	23	modified	modify	VERB
ejpam-3781	44	24	near	near	ADP
ejpam-3781	44	25	module	module	NOUN
ejpam-3781	44	26	.	.	PUNCT
ejpam-3781	44	27	example	example	NOUN
ejpam-3781	45	1	2	2	NUM
ejpam-3781	45	2	.	.	PUNCT
ejpam-3781	46	1	let	let	VERB
ejpam-3781	46	2	n	n	PRON
ejpam-3781	46	3	be	be	AUX
ejpam-3781	46	4	a	a	DET
ejpam-3781	46	5	nontrivial	nontrivial	NOUN
ejpam-3781	46	6	near	near	ADP
ejpam-3781	46	7	ring	ring	NOUN
ejpam-3781	46	8	with	with	ADP
ejpam-3781	46	9	ab	ab	PROPN
ejpam-3781	46	10	=	=	PUNCT
ejpam-3781	46	11	a.	a.	NOUN
ejpam-3781	46	12	let	let	VERB
ejpam-3781	46	13	m	m	VERB
ejpam-3781	46	14	=	=	PUNCT
ejpam-3781	46	15	(	(	PUNCT
ejpam-3781	46	16	n,+	n,+	NUM
ejpam-3781	46	17	)	)	PUNCT
ejpam-3781	46	18	.	.	PUNCT
ejpam-3781	47	1	define	define	VERB
ejpam-3781	47	2	the	the	DET
ejpam-3781	47	3	function	function	NOUN
ejpam-3781	47	4	�	�	PROPN
ejpam-3781	47	5	from	from	ADP
ejpam-3781	47	6	n	n	PROPN
ejpam-3781	47	7	×m	×m	NOUN
ejpam-3781	47	8	into	into	ADP
ejpam-3781	47	9	m	m	PROPN
ejpam-3781	47	10	as	as	ADP
ejpam-3781	47	11	�	�	PROPN
ejpam-3781	47	12	(	(	PUNCT
ejpam-3781	47	13	n	n	CCONJ
ejpam-3781	47	14	,	,	PUNCT
ejpam-3781	47	15	m	m	NOUN
ejpam-3781	47	16	)	)	PUNCT
ejpam-3781	47	17	=	=	SYM
ejpam-3781	48	1	n	n	PRON
ejpam-3781	48	2	�	�	NOUN
ejpam-3781	48	3	m	m	NOUN
ejpam-3781	48	4	=	=	NOUN
ejpam-3781	48	5	m	m	VERB
ejpam-3781	48	6	for	for	ADP
ejpam-3781	48	7	all	all	PRON
ejpam-3781	48	8	n	n	PRON
ejpam-3781	48	9	∈	∈	PROPN
ejpam-3781	48	10	n	n	CCONJ
ejpam-3781	48	11	,	,	PUNCT
ejpam-3781	48	12	m	m	NOUN
ejpam-3781	48	13	∈m	∈m	NOUN
ejpam-3781	48	14	.	.	PUNCT
ejpam-3781	49	1	for	for	ADP
ejpam-3781	49	2	any	any	DET
ejpam-3781	49	3	n	n	CCONJ
ejpam-3781	49	4	,	,	PUNCT
ejpam-3781	49	5	n1	n1	NOUN
ejpam-3781	49	6	,	,	PUNCT
ejpam-3781	49	7	n2	n2	NOUN
ejpam-3781	49	8	∈	∈	PROPN
ejpam-3781	49	9	n	n	PRON
ejpam-3781	49	10	and	and	CCONJ
ejpam-3781	49	11	m	m	PROPN
ejpam-3781	49	12	,	,	PUNCT
ejpam-3781	49	13	m1,m2	m1,m2	PROPN
ejpam-3781	49	14	∈m	∈m	NOUN
ejpam-3781	49	15	,	,	PUNCT
ejpam-3781	49	16	(	(	PUNCT
ejpam-3781	49	17	1	1	NUM
ejpam-3781	49	18	)	)	PUNCT
ejpam-3781	49	19	(	(	PUNCT
ejpam-3781	49	20	n1n2)	n1n2)	PROPN
ejpam-3781	49	21	�	�	PROPN
ejpam-3781	49	22	m	m	NOUN
ejpam-3781	49	23	=	=	ADJ
ejpam-3781	49	24	n1	n1	PROPN
ejpam-3781	49	25	�	�	PROPN
ejpam-3781	49	26	m	m	NOUN
ejpam-3781	49	27	=	=	NOUN
ejpam-3781	49	28	m	m	PROPN
ejpam-3781	49	29	and	and	CCONJ
ejpam-3781	49	30	n1	n1	PROPN
ejpam-3781	49	31	�	�	PROPN
ejpam-3781	49	32	(	(	PUNCT
ejpam-3781	49	33	n2	n2	PROPN
ejpam-3781	49	34	�	�	PROPN
ejpam-3781	49	35	m	m	NOUN
ejpam-3781	49	36	)	)	PUNCT
ejpam-3781	49	37	=	=	SYM
ejpam-3781	49	38	n2	n2	PROPN
ejpam-3781	49	39	�	�	PROPN
ejpam-3781	49	40	m	m	NOUN
ejpam-3781	49	41	=	=	NOUN
ejpam-3781	49	42	m	m	PROPN
ejpam-3781	49	43	(	(	PUNCT
ejpam-3781	49	44	2	2	NUM
ejpam-3781	49	45	)	)	PUNCT
ejpam-3781	49	46	n	n	CCONJ
ejpam-3781	49	47	�	�	PROPN
ejpam-3781	49	48	(	(	PUNCT
ejpam-3781	49	49	m1	m1	PROPN
ejpam-3781	49	50	+	+	NOUN
ejpam-3781	49	51	m2	m2	PROPN
ejpam-3781	49	52	)	)	PUNCT
ejpam-3781	49	53	=	=	PROPN
ejpam-3781	49	54	m1	m1	PROPN
ejpam-3781	49	55	+	+	NUM
ejpam-3781	49	56	m2	m2	PROPN
ejpam-3781	49	57	and	and	CCONJ
ejpam-3781	49	58	n	n	CCONJ
ejpam-3781	49	59	�	�	PROPN
ejpam-3781	49	60	m1	m1	PROPN
ejpam-3781	49	61	+	+	CCONJ
ejpam-3781	49	62	n	n	CCONJ
ejpam-3781	49	63	�	�	NOUN
ejpam-3781	49	64	m2	m2	PROPN
ejpam-3781	49	65	=	=	PROPN
ejpam-3781	49	66	m1	m1	PROPN
ejpam-3781	49	67	+	+	PROPN
ejpam-3781	49	68	m2	m2	PROPN
ejpam-3781	49	69	.	.	PUNCT
ejpam-3781	49	70	therefore	therefore	ADV
ejpam-3781	49	71	(	(	PUNCT
ejpam-3781	49	72	m,+	m,+	PROPN
ejpam-3781	49	73	,	,	PUNCT
ejpam-3781	49	74	�	�	PROPN
ejpam-3781	49	75	)	)	PUNCT
ejpam-3781	49	76	is	be	AUX
ejpam-3781	49	77	a	a	DET
ejpam-3781	49	78	modified	modify	VERB
ejpam-3781	49	79	near	near	ADP
ejpam-3781	49	80	module	module	NOUN
ejpam-3781	49	81	over	over	ADP
ejpam-3781	49	82	n	n	PROPN
ejpam-3781	49	83	.	.	PUNCT
ejpam-3781	50	1	however	however	ADV
ejpam-3781	50	2	(	(	PUNCT
ejpam-3781	50	3	m,+	m,+	PROPN
ejpam-3781	50	4	,	,	PUNCT
ejpam-3781	50	5	�	�	PROPN
ejpam-3781	50	6	)	)	PUNCT
ejpam-3781	50	7	is	be	AUX
ejpam-3781	50	8	not	not	PART
ejpam-3781	50	9	a	a	DET
ejpam-3781	50	10	near	near	ADJ
ejpam-3781	50	11	module	module	NOUN
ejpam-3781	50	12	since	since	SCONJ
ejpam-3781	50	13	(	(	PUNCT
ejpam-3781	50	14	n1	n1	PROPN
ejpam-3781	50	15	+	+	CCONJ
ejpam-3781	50	16	n2)	n2)	NUM
ejpam-3781	50	17	�	�	NOUN
ejpam-3781	50	18	m	m	NOUN
ejpam-3781	50	19	=	=	NOUN
ejpam-3781	50	20	m	m	PROPN
ejpam-3781	50	21	and	and	CCONJ
ejpam-3781	50	22	n1	n1	PROPN
ejpam-3781	50	23	�	�	PROPN
ejpam-3781	50	24	m+	m+	NUM
ejpam-3781	50	25	n2	n2	PROPN
ejpam-3781	50	26	�	�	PROPN
ejpam-3781	50	27	m	m	PROPN
ejpam-3781	50	28	=	=	SYM
ejpam-3781	50	29	m+m	m+m	PROPN
ejpam-3781	50	30	.	.	PUNCT
ejpam-3781	51	1	a.v	a.v	PROPN
ejpam-3781	51	2	.	.	PROPN
ejpam-3781	51	3	ramakrishna	ramakrishna	PROPN
ejpam-3781	51	4	,	,	PUNCT
ejpam-3781	51	5	t.v.n	t.v.n	PROPN
ejpam-3781	51	6	.	.	PUNCT
ejpam-3781	52	1	prasanna	prasanna	PROPN
ejpam-3781	52	2	,	,	PUNCT
ejpam-3781	52	3	d.v	d.v	PROPN
ejpam-3781	52	4	.	.	PROPN
ejpam-3781	52	5	lakshmi	lakshmi	PROPN
ejpam-3781	52	6	/	/	SYM
ejpam-3781	52	7	eur	eur	PROPN
ejpam-3781	52	8	.	.	PUNCT
ejpam-3781	53	1	j.	j.	PROPN
ejpam-3781	53	2	pure	pure	PROPN
ejpam-3781	53	3	appl	appl	PROPN
ejpam-3781	53	4	.	.	PROPN
ejpam-3781	53	5	math	math	PROPN
ejpam-3781	53	6	,	,	PUNCT
ejpam-3781	53	7	14	14	NUM
ejpam-3781	53	8	(	(	PUNCT
ejpam-3781	53	9	1	1	NUM
ejpam-3781	53	10	)	)	PUNCT
ejpam-3781	53	11	(	(	PUNCT
ejpam-3781	53	12	2021	2021	NUM
ejpam-3781	53	13	)	)	PUNCT
ejpam-3781	53	14	,	,	PUNCT
ejpam-3781	53	15	126	126	NUM
ejpam-3781	53	16	-	-	SYM
ejpam-3781	53	17	134	134	NUM
ejpam-3781	53	18	128	128	NUM
ejpam-3781	53	19	example	example	NOUN
ejpam-3781	53	20	3	3	NUM
ejpam-3781	53	21	.	.	PUNCT
ejpam-3781	54	1	let	let	VERB
ejpam-3781	54	2	r	r	NOUN
ejpam-3781	54	3	=	=	PUNCT
ejpam-3781	54	4	(	(	PUNCT
ejpam-3781	54	5	r,+	r,+	NUM
ejpam-3781	54	6	,	,	PUNCT
ejpam-3781	54	7	·	·	PUNCT
ejpam-3781	54	8	)	)	PUNCT
ejpam-3781	54	9	.	.	PUNCT
ejpam-3781	55	1	define	define	VERB
ejpam-3781	55	2	φ	φ	NOUN
ejpam-3781	55	3	:	:	PUNCT
ejpam-3781	55	4	r→	r→	PROPN
ejpam-3781	55	5	r	r	NOUN
ejpam-3781	55	6	by	by	ADP
ejpam-3781	55	7	φ(r	φ(r	ADJ
ejpam-3781	55	8	)	)	PUNCT
ejpam-3781	55	9	=	=	SYM
ejpam-3781	55	10	r2	r2	NOUN
ejpam-3781	55	11	.	.	PUNCT
ejpam-3781	56	1	define	define	VERB
ejpam-3781	56	2	‘	'	PUNCT
ejpam-3781	56	3	�	�	NOUN
ejpam-3781	56	4	’	'	PUNCT
ejpam-3781	56	5	:	:	PUNCT
ejpam-3781	56	6	r×	r×	NOUN
ejpam-3781	56	7	r	r	NOUN
ejpam-3781	56	8	into	into	ADP
ejpam-3781	56	9	r	r	NOUN
ejpam-3781	56	10	as	as	ADP
ejpam-3781	56	11	�	�	PROPN
ejpam-3781	56	12	(	(	PUNCT
ejpam-3781	56	13	r	r	NOUN
ejpam-3781	56	14	,	,	PUNCT
ejpam-3781	56	15	m	m	NOUN
ejpam-3781	56	16	)	)	PUNCT
ejpam-3781	56	17	=	=	SYM
ejpam-3781	56	18	r	r	NOUN
ejpam-3781	56	19	�	�	PROPN
ejpam-3781	56	20	m	m	NOUN
ejpam-3781	56	21	=	=	NOUN
ejpam-3781	56	22	r2	r2	PROPN
ejpam-3781	56	23	m	m	VERB
ejpam-3781	56	24	for	for	ADP
ejpam-3781	56	25	r	r	NOUN
ejpam-3781	56	26	,	,	PUNCT
ejpam-3781	56	27	m	m	PROPN
ejpam-3781	56	28	∈	∈	PROPN
ejpam-3781	56	29	r.	r.	NOUN
ejpam-3781	56	30	then	then	ADV
ejpam-3781	56	31	(	(	PUNCT
ejpam-3781	56	32	r,+	r,+	NUM
ejpam-3781	56	33	)	)	PUNCT
ejpam-3781	56	34	is	be	AUX
ejpam-3781	56	35	a	a	DET
ejpam-3781	56	36	modified	modify	VERB
ejpam-3781	56	37	near	near	ADP
ejpam-3781	56	38	module	module	NOUN
ejpam-3781	56	39	over	over	ADP
ejpam-3781	56	40	the	the	DET
ejpam-3781	56	41	near	near	ADJ
ejpam-3781	56	42	ring	ring	NOUN
ejpam-3781	56	43	(	(	PUNCT
ejpam-3781	56	44	r,+	r,+	NUM
ejpam-3781	56	45	,	,	PUNCT
ejpam-3781	56	46	·	·	PUNCT
ejpam-3781	56	47	)	)	PUNCT
ejpam-3781	56	48	but	but	CCONJ
ejpam-3781	56	49	not	not	PART
ejpam-3781	56	50	a	a	DET
ejpam-3781	56	51	near	near	ADJ
ejpam-3781	56	52	module	module	NOUN
ejpam-3781	56	53	.	.	PUNCT
ejpam-3781	57	1	that	that	PRON
ejpam-3781	57	2	(	(	PUNCT
ejpam-3781	57	3	r,+	r,+	NUM
ejpam-3781	57	4	)	)	PUNCT
ejpam-3781	57	5	is	be	AUX
ejpam-3781	57	6	a	a	DET
ejpam-3781	57	7	modified	modify	VERB
ejpam-3781	57	8	near	near	ADP
ejpam-3781	57	9	module	module	NOUN
ejpam-3781	57	10	can	can	AUX
ejpam-3781	57	11	be	be	AUX
ejpam-3781	57	12	verified	verify	VERB
ejpam-3781	57	13	easily	easily	ADV
ejpam-3781	57	14	.	.	PUNCT
ejpam-3781	58	1	however	however	ADV
ejpam-3781	58	2	r	r	NOUN
ejpam-3781	58	3	is	be	AUX
ejpam-3781	58	4	not	not	PART
ejpam-3781	58	5	a	a	DET
ejpam-3781	58	6	near	near	ADJ
ejpam-3781	58	7	module	module	NOUN
ejpam-3781	58	8	as	as	SCONJ
ejpam-3781	58	9	is	be	AUX
ejpam-3781	58	10	evident	evident	ADJ
ejpam-3781	58	11	from	from	ADP
ejpam-3781	58	12	the	the	DET
ejpam-3781	58	13	following	following	NOUN
ejpam-3781	58	14	:	:	PUNCT
ejpam-3781	58	15	take	take	VERB
ejpam-3781	58	16	r1	r1	NOUN
ejpam-3781	58	17	=	=	SYM
ejpam-3781	58	18	1	1	NUM
ejpam-3781	58	19	,	,	PUNCT
ejpam-3781	58	20	r2	r2	PROPN
ejpam-3781	58	21	=	=	SYM
ejpam-3781	59	1	1,m	1,m	PROPN
ejpam-3781	59	2	=	=	SYM
ejpam-3781	60	1	2	2	X
ejpam-3781	60	2	.	.	PUNCT
ejpam-3781	60	3	then	then	ADV
ejpam-3781	60	4	(	(	PUNCT
ejpam-3781	60	5	r1	r1	PROPN
ejpam-3781	60	6	+	+	CCONJ
ejpam-3781	60	7	r2)	r2)	NOUN
ejpam-3781	60	8	�	�	NOUN
ejpam-3781	60	9	m	m	NOUN
ejpam-3781	60	10	=	=	PUNCT
ejpam-3781	60	11	(	(	PUNCT
ejpam-3781	60	12	1	1	NUM
ejpam-3781	60	13	+	+	NUM
ejpam-3781	60	14	1	1	NUM
ejpam-3781	60	15	)	)	PUNCT
ejpam-3781	60	16	�	�	PROPN
ejpam-3781	60	17	2	2	NUM
ejpam-3781	60	18	=	=	SYM
ejpam-3781	60	19	2	2	NUM
ejpam-3781	60	20	�	�	PROPN
ejpam-3781	60	21	2	2	NUM
ejpam-3781	60	22	=	=	SYM
ejpam-3781	60	23	222	222	NUM
ejpam-3781	60	24	=	=	SYM
ejpam-3781	60	25	8	8	NUM
ejpam-3781	60	26	and	and	CCONJ
ejpam-3781	60	27	r1	r1	PROPN
ejpam-3781	60	28	�	�	PROPN
ejpam-3781	60	29	m+	m+	NUM
ejpam-3781	60	30	r2	r2	PROPN
ejpam-3781	60	31	�	�	PROPN
ejpam-3781	60	32	m	m	NOUN
ejpam-3781	60	33	=	=	ADJ
ejpam-3781	60	34	1	1	NUM
ejpam-3781	60	35	�	�	PROPN
ejpam-3781	60	36	2	2	NUM
ejpam-3781	60	37	+	+	SYM
ejpam-3781	60	38	1	1	NUM
ejpam-3781	60	39	�	�	PROPN
ejpam-3781	60	40	2	2	NUM
ejpam-3781	60	41	=	=	SYM
ejpam-3781	60	42	122	122	NUM
ejpam-3781	61	1	+	+	NUM
ejpam-3781	61	2	122	122	NUM
ejpam-3781	61	3	=	=	SYM
ejpam-3781	61	4	2	2	NUM
ejpam-3781	61	5	+	+	NUM
ejpam-3781	61	6	2	2	NUM
ejpam-3781	61	7	=	=	SYM
ejpam-3781	61	8	4	4	NUM
ejpam-3781	61	9	.	.	X
ejpam-3781	61	10	infact	infact	VERB
ejpam-3781	61	11	the	the	DET
ejpam-3781	61	12	above	above	ADJ
ejpam-3781	61	13	example	example	NOUN
ejpam-3781	61	14	is	be	AUX
ejpam-3781	61	15	a	a	DET
ejpam-3781	61	16	special	special	ADJ
ejpam-3781	61	17	case	case	NOUN
ejpam-3781	61	18	of	of	ADP
ejpam-3781	61	19	the	the	DET
ejpam-3781	61	20	following	follow	VERB
ejpam-3781	61	21	theorem	theorem	NOUN
ejpam-3781	61	22	:	:	PUNCT
ejpam-3781	61	23	theorem	theorem	NOUN
ejpam-3781	61	24	1	1	X
ejpam-3781	61	25	.	.	PUNCT
ejpam-3781	62	1	let	let	AUX
ejpam-3781	62	2	(	(	PUNCT
ejpam-3781	62	3	r,+	r,+	NUM
ejpam-3781	62	4	,	,	PUNCT
ejpam-3781	62	5	·	·	PUNCT
ejpam-3781	62	6	)	)	PUNCT
ejpam-3781	62	7	and	and	CCONJ
ejpam-3781	62	8	(	(	PUNCT
ejpam-3781	62	9	s,+1	s,+1	ADJ
ejpam-3781	62	10	,	,	PUNCT
ejpam-3781	62	11	·	·	PUNCT
ejpam-3781	62	12	1	1	X
ejpam-3781	62	13	)	)	PUNCT
ejpam-3781	62	14	be	be	AUX
ejpam-3781	62	15	near	near	ADP
ejpam-3781	62	16	rings	ring	NOUN
ejpam-3781	62	17	and	and	CCONJ
ejpam-3781	62	18	let	let	VERB
ejpam-3781	62	19	φ	φ	NOUN
ejpam-3781	62	20	:	:	PUNCT
ejpam-3781	62	21	r	r	X
ejpam-3781	62	22	→	→	SYM
ejpam-3781	62	23	s	s	AUX
ejpam-3781	62	24	be	be	AUX
ejpam-3781	62	25	a	a	DET
ejpam-3781	62	26	mapping	mapping	NOUN
ejpam-3781	62	27	such	such	ADJ
ejpam-3781	62	28	that	that	SCONJ
ejpam-3781	62	29	φ(r1	φ(r1	ADV
ejpam-3781	62	30	·	·	PUNCT
ejpam-3781	62	31	r2	r2	PROPN
ejpam-3781	62	32	)	)	PUNCT
ejpam-3781	63	1	=	=	SYM
ejpam-3781	63	2	φ(r1	φ(r1	ADV
ejpam-3781	63	3	)	)	PUNCT
ejpam-3781	63	4	·	·	SYM
ejpam-3781	63	5	1	1	NUM
ejpam-3781	63	6	φ(r2	φ(r2	NOUN
ejpam-3781	63	7	)	)	PUNCT
ejpam-3781	63	8	for	for	ADP
ejpam-3781	63	9	all	all	DET
ejpam-3781	63	10	r1	r1	NOUN
ejpam-3781	63	11	,	,	PUNCT
ejpam-3781	63	12	r2	r2	PROPN
ejpam-3781	63	13	∈	∈	PROPN
ejpam-3781	63	14	r.	r.	PROPN
ejpam-3781	63	15	let	let	VERB
ejpam-3781	63	16	(	(	PUNCT
ejpam-3781	63	17	m,⊕	m,⊕	ADJ
ejpam-3781	63	18	,	,	PUNCT
ejpam-3781	63	19	�	�	PROPN
ejpam-3781	63	20	)	)	PUNCT
ejpam-3781	63	21	be	be	VERB
ejpam-3781	63	22	a	a	DET
ejpam-3781	63	23	left	left	ADJ
ejpam-3781	63	24	s	s	NOUN
ejpam-3781	63	25	-	-	NOUN
ejpam-3781	63	26	module	module	NOUN
ejpam-3781	63	27	.	.	PUNCT
ejpam-3781	64	1	therefore	therefore	ADV
ejpam-3781	64	2	(	(	PUNCT
ejpam-3781	64	3	m,⊕,	m,⊕,	X
ejpam-3781	64	4	�	�	PROPN
ejpam-3781	64	5	1	1	NUM
ejpam-3781	64	6	)	)	PUNCT
ejpam-3781	64	7	is	be	AUX
ejpam-3781	64	8	a	a	DET
ejpam-3781	64	9	modified	modify	VERB
ejpam-3781	64	10	near	near	ADP
ejpam-3781	64	11	module	module	NOUN
ejpam-3781	64	12	over	over	ADP
ejpam-3781	64	13	the	the	DET
ejpam-3781	64	14	near	near	ADJ
ejpam-3781	64	15	ring	ring	NOUN
ejpam-3781	64	16	(	(	PUNCT
ejpam-3781	64	17	r,+	r,+	NUM
ejpam-3781	64	18	,	,	PUNCT
ejpam-3781	64	19	·	·	PUNCT
ejpam-3781	64	20	)	)	PUNCT
ejpam-3781	64	21	when	when	SCONJ
ejpam-3781	64	22	�	�	X
ejpam-3781	64	23	1	1	NUM
ejpam-3781	64	24	is	be	AUX
ejpam-3781	64	25	defined	define	VERB
ejpam-3781	64	26	by	by	ADP
ejpam-3781	64	27	r	r	PROPN
ejpam-3781	64	28	�	�	PROPN
ejpam-3781	64	29	1	1	NUM
ejpam-3781	64	30	m	m	PROPN
ejpam-3781	64	31	=	=	PROPN
ejpam-3781	64	32	φ(r)	φ(r)	NOUN
ejpam-3781	64	33	�	�	NOUN
ejpam-3781	64	34	m	m	VERB
ejpam-3781	64	35	for	for	ADP
ejpam-3781	64	36	all	all	DET
ejpam-3781	64	37	r	r	NOUN
ejpam-3781	64	38	∈	∈	NOUN
ejpam-3781	64	39	r	r	NOUN
ejpam-3781	64	40	and	and	CCONJ
ejpam-3781	64	41	m	m	NOUN
ejpam-3781	64	42	∈m	∈m	NOUN
ejpam-3781	64	43	.	.	PUNCT
ejpam-3781	65	1	proof	proof	NOUN
ejpam-3781	65	2	.	.	PUNCT
ejpam-3781	66	1	since	since	SCONJ
ejpam-3781	66	2	(	(	PUNCT
ejpam-3781	66	3	m,⊕	m,⊕	NOUN
ejpam-3781	66	4	,	,	PUNCT
ejpam-3781	66	5	�	�	PROPN
ejpam-3781	66	6	)	)	PUNCT
ejpam-3781	66	7	is	be	AUX
ejpam-3781	66	8	a	a	DET
ejpam-3781	66	9	left	left	ADJ
ejpam-3781	66	10	s	s	NOUN
ejpam-3781	66	11	-	-	NOUN
ejpam-3781	66	12	module	module	NOUN
ejpam-3781	66	13	,	,	PUNCT
ejpam-3781	66	14	(	(	PUNCT
ejpam-3781	66	15	i	i	NOUN
ejpam-3781	66	16	)	)	PUNCT
ejpam-3781	67	1	s	s	PROPN
ejpam-3781	67	2	�	�	PROPN
ejpam-3781	67	3	(	(	PUNCT
ejpam-3781	67	4	m1	m1	PROPN
ejpam-3781	67	5	⊕m2	⊕m2	NUM
ejpam-3781	67	6	)	)	PUNCT
ejpam-3781	67	7	=	=	SYM
ejpam-3781	68	1	s	s	PROPN
ejpam-3781	68	2	�	�	PROPN
ejpam-3781	68	3	m1	m1	PROPN
ejpam-3781	68	4	⊕	⊕	PROPN
ejpam-3781	68	5	s	s	PROPN
ejpam-3781	68	6	�	�	PROPN
ejpam-3781	68	7	m2	m2	PROPN
ejpam-3781	68	8	;	;	PUNCT
ejpam-3781	68	9	(	(	PUNCT
ejpam-3781	68	10	ii	ii	NOUN
ejpam-3781	68	11	)	)	PUNCT
ejpam-3781	68	12	(	(	PUNCT
ejpam-3781	68	13	s1	s1	PROPN
ejpam-3781	68	14	+1	+1	PROPN
ejpam-3781	68	15	s2)	s2)	PROPN
ejpam-3781	68	16	�	�	PROPN
ejpam-3781	68	17	m	m	NOUN
ejpam-3781	68	18	=	=	PROPN
ejpam-3781	68	19	s1	s1	PROPN
ejpam-3781	68	20	�	�	PROPN
ejpam-3781	68	21	m⊕	m⊕	VERB
ejpam-3781	68	22	s2	s2	NOUN
ejpam-3781	68	23	�	�	NOUN
ejpam-3781	68	24	m	m	PROPN
ejpam-3781	68	25	;	;	PUNCT
ejpam-3781	68	26	(	(	PUNCT
ejpam-3781	68	27	iii	iii	X
ejpam-3781	68	28	)	)	PUNCT
ejpam-3781	68	29	s1	s1	PROPN
ejpam-3781	68	30	�	�	PROPN
ejpam-3781	68	31	(	(	PUNCT
ejpam-3781	68	32	s2	s2	PROPN
ejpam-3781	68	33	�	�	PROPN
ejpam-3781	68	34	m	m	NOUN
ejpam-3781	68	35	)	)	PUNCT
ejpam-3781	68	36	=	=	PRON
ejpam-3781	69	1	(	(	PUNCT
ejpam-3781	69	2	s1	s1	PROPN
ejpam-3781	69	3	·	·	SYM
ejpam-3781	69	4	1	1	NUM
ejpam-3781	69	5	s2)	s2)	PROPN
ejpam-3781	69	6	�	�	PROPN
ejpam-3781	69	7	m	m	VERB
ejpam-3781	69	8	for	for	ADP
ejpam-3781	69	9	all	all	DET
ejpam-3781	69	10	s	s	PROPN
ejpam-3781	69	11	,	,	PUNCT
ejpam-3781	69	12	s1	s1	NOUN
ejpam-3781	69	13	,	,	PUNCT
ejpam-3781	69	14	s2	s2	NOUN
ejpam-3781	69	15	∈	∈	PROPN
ejpam-3781	69	16	s	s	PROPN
ejpam-3781	69	17	and	and	CCONJ
ejpam-3781	69	18	m	m	PROPN
ejpam-3781	69	19	,	,	PUNCT
ejpam-3781	69	20	m1,m2	m1,m2	PROPN
ejpam-3781	69	21	∈m	∈m	NOUN
ejpam-3781	69	22	.	.	PUNCT
ejpam-3781	70	1	for	for	ADP
ejpam-3781	70	2	any	any	DET
ejpam-3781	70	3	r	r	NOUN
ejpam-3781	70	4	,	,	PUNCT
ejpam-3781	70	5	r1	r1	NOUN
ejpam-3781	70	6	,	,	PUNCT
ejpam-3781	70	7	r2	r2	PROPN
ejpam-3781	70	8	∈	∈	PROPN
ejpam-3781	70	9	r	r	NOUN
ejpam-3781	70	10	and	and	CCONJ
ejpam-3781	70	11	m	m	PROPN
ejpam-3781	70	12	,	,	PUNCT
ejpam-3781	70	13	m1,m2	m1,m2	PROPN
ejpam-3781	70	14	∈m	∈m	NOUN
ejpam-3781	70	15	,	,	PUNCT
ejpam-3781	70	16	(	(	PUNCT
ejpam-3781	70	17	1	1	X
ejpam-3781	70	18	)	)	PUNCT
ejpam-3781	70	19	r1	r1	PROPN
ejpam-3781	70	20	�	�	PROPN
ejpam-3781	70	21	1	1	NUM
ejpam-3781	70	22	(	(	PUNCT
ejpam-3781	70	23	r2	r2	PROPN
ejpam-3781	70	24	�	�	PROPN
ejpam-3781	70	25	1	1	NUM
ejpam-3781	70	26	m	m	NOUN
ejpam-3781	70	27	)	)	PUNCT
ejpam-3781	70	28	=	=	SYM
ejpam-3781	70	29	φ(r1	φ(r1	ADV
ejpam-3781	70	30	)	)	PUNCT
ejpam-3781	70	31	�	�	PROPN
ejpam-3781	70	32	(	(	PUNCT
ejpam-3781	70	33	r2	r2	PROPN
ejpam-3781	70	34	�	�	PROPN
ejpam-3781	70	35	1	1	NUM
ejpam-3781	70	36	m	m	NOUN
ejpam-3781	70	37	)	)	PUNCT
ejpam-3781	70	38	=	=	SYM
ejpam-3781	70	39	φ(r1	φ(r1	ADV
ejpam-3781	70	40	)	)	PUNCT
ejpam-3781	70	41	�	�	PROPN
ejpam-3781	71	1	[	[	X
ejpam-3781	71	2	φ(r2)	φ(r2)	NOUN
ejpam-3781	71	3	�	�	PROPN
ejpam-3781	71	4	m	m	NOUN
ejpam-3781	71	5	]	]	X
ejpam-3781	72	1	=	=	PUNCT
ejpam-3781	73	1	[	[	X
ejpam-3781	73	2	φ(r1	φ(r1	ADV
ejpam-3781	73	3	)	)	PUNCT
ejpam-3781	73	4	·	·	SYM
ejpam-3781	73	5	1	1	NUM
ejpam-3781	73	6	φ(r2)]	φ(r2)]	NUM
ejpam-3781	73	7	�	�	NOUN
ejpam-3781	73	8	m	m	NOUN
ejpam-3781	73	9	=	=	NOUN
ejpam-3781	73	10	φ(r1r2)	φ(r1r2)	NOUN
ejpam-3781	73	11	�	�	NOUN
ejpam-3781	73	12	m	m	NOUN
ejpam-3781	73	13	=	=	SYM
ejpam-3781	73	14	(	(	PUNCT
ejpam-3781	73	15	r1r2)	r1r2)	NOUN
ejpam-3781	73	16	�	�	NOUN
ejpam-3781	73	17	1	1	NUM
ejpam-3781	73	18	m	m	NOUN
ejpam-3781	73	19	and	and	CCONJ
ejpam-3781	73	20	(	(	PUNCT
ejpam-3781	73	21	2	2	X
ejpam-3781	73	22	)	)	PUNCT
ejpam-3781	73	23	r	r	NOUN
ejpam-3781	73	24	�	�	PROPN
ejpam-3781	73	25	1	1	NUM
ejpam-3781	73	26	(	(	PUNCT
ejpam-3781	73	27	m1	m1	PROPN
ejpam-3781	73	28	⊕m2	⊕m2	NUM
ejpam-3781	73	29	)	)	PUNCT
ejpam-3781	73	30	=	=	SYM
ejpam-3781	73	31	φ(r	φ(r	ADJ
ejpam-3781	73	32	)	)	PUNCT
ejpam-3781	73	33	�	�	PROPN
ejpam-3781	74	1	[	[	X
ejpam-3781	74	2	m1	m1	PROPN
ejpam-3781	74	3	⊕m2	⊕m2	NUM
ejpam-3781	74	4	]	]	X
ejpam-3781	74	5	=	=	SYM
ejpam-3781	74	6	φ(r)	φ(r)	PROPN
ejpam-3781	74	7	�	�	PROPN
ejpam-3781	74	8	m1	m1	PROPN
ejpam-3781	74	9	⊕	⊕	PROPN
ejpam-3781	74	10	φ(r)	φ(r)	PROPN
ejpam-3781	74	11	�	�	PROPN
ejpam-3781	74	12	m2	m2	PROPN
ejpam-3781	74	13	=	=	PROPN
ejpam-3781	74	14	r	r	NOUN
ejpam-3781	74	15	�	�	PROPN
ejpam-3781	74	16	1	1	NUM
ejpam-3781	74	17	m1	m1	PROPN
ejpam-3781	74	18	⊕	⊕	PROPN
ejpam-3781	74	19	r	r	NOUN
ejpam-3781	74	20	�	�	PROPN
ejpam-3781	74	21	1	1	NUM
ejpam-3781	74	22	m2	m2	PROPN
ejpam-3781	74	23	.	.	PUNCT
ejpam-3781	75	1	therefore	therefore	ADV
ejpam-3781	75	2	(	(	PUNCT
ejpam-3781	75	3	m,+	m,+	PROPN
ejpam-3781	75	4	,	,	PUNCT
ejpam-3781	75	5	·	·	PUNCT
ejpam-3781	75	6	)	)	PUNCT
ejpam-3781	75	7	is	be	AUX
ejpam-3781	75	8	a	a	DET
ejpam-3781	75	9	modified	modify	VERB
ejpam-3781	75	10	near	near	ADP
ejpam-3781	75	11	module	module	NOUN
ejpam-3781	75	12	over	over	ADP
ejpam-3781	75	13	(	(	PUNCT
ejpam-3781	75	14	r,+	r,+	NUM
ejpam-3781	75	15	,	,	PUNCT
ejpam-3781	75	16	·	·	PUNCT
ejpam-3781	75	17	)	)	PUNCT
ejpam-3781	75	18	.	.	PUNCT
ejpam-3781	76	1	definition	definition	NOUN
ejpam-3781	76	2	4	4	X
ejpam-3781	76	3	.	.	PUNCT
ejpam-3781	77	1	let	let	AUX
ejpam-3781	77	2	(	(	PUNCT
ejpam-3781	77	3	m,+	m,+	INTJ
ejpam-3781	77	4	,	,	PUNCT
ejpam-3781	77	5	·	·	PUNCT
ejpam-3781	77	6	)	)	PUNCT
ejpam-3781	77	7	be	be	AUX
ejpam-3781	77	8	a	a	DET
ejpam-3781	77	9	modified	modify	VERB
ejpam-3781	77	10	near	near	ADP
ejpam-3781	77	11	module	module	NOUN
ejpam-3781	77	12	over	over	ADP
ejpam-3781	77	13	n	n	PROPN
ejpam-3781	77	14	.	.	PUNCT
ejpam-3781	78	1	a	a	DET
ejpam-3781	78	2	normal	normal	ADJ
ejpam-3781	78	3	subgroup	subgroup	NOUN
ejpam-3781	78	4	i	i	PRON
ejpam-3781	78	5	of	of	ADP
ejpam-3781	78	6	m	m	PROPN
ejpam-3781	78	7	is	be	AUX
ejpam-3781	78	8	called	call	VERB
ejpam-3781	78	9	an	an	DET
ejpam-3781	78	10	ideal	ideal	NOUN
ejpam-3781	78	11	of	of	ADP
ejpam-3781	78	12	m	m	PRON
ejpam-3781	78	13	if	if	SCONJ
ejpam-3781	78	14	n(m+	n(m+	VERB
ejpam-3781	78	15	i)−	i)−	ADJ
ejpam-3781	78	16	nm	nm	ADJ
ejpam-3781	78	17	∈	∈	NOUN
ejpam-3781	78	18	i	i	PRON
ejpam-3781	78	19	for	for	ADP
ejpam-3781	78	20	all	all	PRON
ejpam-3781	78	21	n	n	PRON
ejpam-3781	78	22	∈	∈	PROPN
ejpam-3781	78	23	n	n	CCONJ
ejpam-3781	78	24	,	,	PUNCT
ejpam-3781	78	25	i	i	PRON
ejpam-3781	78	26	∈	∈	VERB
ejpam-3781	79	1	i	i	PRON
ejpam-3781	79	2	and	and	CCONJ
ejpam-3781	79	3	m	m	VERB
ejpam-3781	79	4	∈m	∈m	NOUN
ejpam-3781	79	5	.	.	PUNCT
ejpam-3781	80	1	definition	definition	NOUN
ejpam-3781	80	2	5	5	NUM
ejpam-3781	80	3	.	.	PUNCT
ejpam-3781	81	1	let	let	VERB
ejpam-3781	81	2	(	(	PUNCT
ejpam-3781	81	3	m1,+1	m1,+1	VERB
ejpam-3781	81	4	,	,	PUNCT
ejpam-3781	81	5	·	·	PUNCT
ejpam-3781	81	6	1	1	NUM
ejpam-3781	81	7	)	)	PUNCT
ejpam-3781	81	8	and	and	CCONJ
ejpam-3781	81	9	(	(	PUNCT
ejpam-3781	81	10	m2,+2	m2,+2	PROPN
ejpam-3781	81	11	,	,	PUNCT
ejpam-3781	81	12	·	·	PUNCT
ejpam-3781	81	13	2	2	X
ejpam-3781	81	14	)	)	PUNCT
ejpam-3781	81	15	be	be	AUX
ejpam-3781	81	16	modified	modify	VERB
ejpam-3781	81	17	near	near	ADP
ejpam-3781	81	18	modules	module	NOUN
ejpam-3781	81	19	over	over	ADP
ejpam-3781	81	20	n	n	PROPN
ejpam-3781	81	21	.	.	PUNCT
ejpam-3781	82	1	a	a	DET
ejpam-3781	82	2	mapping	mapping	NOUN
ejpam-3781	82	3	φ	φ	NOUN
ejpam-3781	82	4	:	:	PUNCT
ejpam-3781	82	5	m1	m1	NOUN
ejpam-3781	82	6	→m2	→m2	PUNCT
ejpam-3781	82	7	is	be	AUX
ejpam-3781	82	8	called	call	VERB
ejpam-3781	82	9	a	a	DET
ejpam-3781	82	10	modified	modify	VERB
ejpam-3781	82	11	near	near	ADP
ejpam-3781	82	12	module	module	NOUN
ejpam-3781	82	13	homomorphism	homomorphism	NOUN
ejpam-3781	82	14	if	if	SCONJ
ejpam-3781	82	15	(	(	PUNCT
ejpam-3781	82	16	i	i	NOUN
ejpam-3781	82	17	)	)	PUNCT
ejpam-3781	82	18	φ(m+1	φ(m+1	PROPN
ejpam-3781	82	19	m	m	VERB
ejpam-3781	82	20	′	′	NUM
ejpam-3781	82	21	)	)	PUNCT
ejpam-3781	82	22	=	=	SYM
ejpam-3781	82	23	φ(m	φ(m	NOUN
ejpam-3781	82	24	)	)	PUNCT
ejpam-3781	82	25	+2	+2	PROPN
ejpam-3781	82	26	φ(m′	φ(m′	NOUN
ejpam-3781	82	27	)	)	PUNCT
ejpam-3781	82	28	;	;	PUNCT
ejpam-3781	82	29	(	(	PUNCT
ejpam-3781	82	30	ii	ii	NOUN
ejpam-3781	82	31	)	)	PUNCT
ejpam-3781	82	32	φ(n	φ(n	PROPN
ejpam-3781	82	33	·	·	SYM
ejpam-3781	82	34	1	1	NUM
ejpam-3781	82	35	m	m	NOUN
ejpam-3781	82	36	)	)	PUNCT
ejpam-3781	82	37	=	=	SYM
ejpam-3781	83	1	n	n	PRON
ejpam-3781	83	2	·	·	SYM
ejpam-3781	83	3	2	2	NUM
ejpam-3781	83	4	φ(m	φ(m	NOUN
ejpam-3781	83	5	)	)	PUNCT
ejpam-3781	83	6	for	for	ADP
ejpam-3781	83	7	all	all	DET
ejpam-3781	83	8	m	m	PROPN
ejpam-3781	83	9	,	,	PUNCT
ejpam-3781	83	10	m′	m′	NOUN
ejpam-3781	83	11	∈m1	∈m1	ADJ
ejpam-3781	83	12	and	and	CCONJ
ejpam-3781	83	13	n	n	PRON
ejpam-3781	83	14	∈	∈	PROPN
ejpam-3781	83	15	n	n	NOUN
ejpam-3781	83	16	.	.	PUNCT
ejpam-3781	84	1	the	the	DET
ejpam-3781	84	2	proofs	proof	NOUN
ejpam-3781	84	3	of	of	ADP
ejpam-3781	84	4	the	the	DET
ejpam-3781	84	5	following	follow	VERB
ejpam-3781	84	6	theorems	theorem	NOUN
ejpam-3781	84	7	are	be	AUX
ejpam-3781	84	8	similar	similar	ADJ
ejpam-3781	84	9	to	to	ADP
ejpam-3781	84	10	those	those	PRON
ejpam-3781	84	11	of	of	ADP
ejpam-3781	84	12	their	their	PRON
ejpam-3781	84	13	counterparts	counterpart	NOUN
ejpam-3781	84	14	in	in	ADP
ejpam-3781	84	15	near	near	ADJ
ejpam-3781	84	16	ring	ring	NOUN
ejpam-3781	84	17	theory	theory	NOUN
ejpam-3781	84	18	[	[	X
ejpam-3781	84	19	5	5	NUM
ejpam-3781	84	20	]	]	PUNCT
ejpam-3781	84	21	,	,	PUNCT
ejpam-3781	84	22	hence	hence	ADV
ejpam-3781	84	23	omitted	omit	VERB
ejpam-3781	84	24	.	.	PUNCT
ejpam-3781	85	1	a.v	a.v	PROPN
ejpam-3781	85	2	.	.	PROPN
ejpam-3781	85	3	ramakrishna	ramakrishna	PROPN
ejpam-3781	85	4	,	,	PUNCT
ejpam-3781	85	5	t.v.n	t.v.n	PROPN
ejpam-3781	85	6	.	.	PUNCT
ejpam-3781	86	1	prasanna	prasanna	PROPN
ejpam-3781	86	2	,	,	PUNCT
ejpam-3781	86	3	d.v	d.v	PROPN
ejpam-3781	86	4	.	.	PROPN
ejpam-3781	86	5	lakshmi	lakshmi	PROPN
ejpam-3781	86	6	/	/	SYM
ejpam-3781	86	7	eur	eur	PROPN
ejpam-3781	86	8	.	.	PUNCT
ejpam-3781	87	1	j.	j.	PROPN
ejpam-3781	87	2	pure	pure	PROPN
ejpam-3781	87	3	appl	appl	PROPN
ejpam-3781	87	4	.	.	PROPN
ejpam-3781	87	5	math	math	PROPN
ejpam-3781	87	6	,	,	PUNCT
ejpam-3781	87	7	14	14	NUM
ejpam-3781	87	8	(	(	PUNCT
ejpam-3781	87	9	1	1	NUM
ejpam-3781	87	10	)	)	PUNCT
ejpam-3781	87	11	(	(	PUNCT
ejpam-3781	87	12	2021	2021	NUM
ejpam-3781	87	13	)	)	PUNCT
ejpam-3781	87	14	,	,	PUNCT
ejpam-3781	87	15	126	126	NUM
ejpam-3781	87	16	-	-	SYM
ejpam-3781	87	17	134	134	NUM
ejpam-3781	87	18	129	129	NUM
ejpam-3781	87	19	theorem	theorem	NOUN
ejpam-3781	87	20	2	2	NUM
ejpam-3781	87	21	.	.	PUNCT
ejpam-3781	88	1	let	let	VERB
ejpam-3781	88	2	m1,m2	m1,m2	PROPN
ejpam-3781	88	3	be	be	AUX
ejpam-3781	88	4	modified	modify	VERB
ejpam-3781	88	5	near	near	ADP
ejpam-3781	88	6	modules	module	NOUN
ejpam-3781	88	7	over	over	ADP
ejpam-3781	88	8	n	n	NOUN
ejpam-3781	88	9	and	and	CCONJ
ejpam-3781	88	10	let	let	VERB
ejpam-3781	88	11	φ	φ	NOUN
ejpam-3781	88	12	:	:	PUNCT
ejpam-3781	88	13	m1	m1	PROPN
ejpam-3781	88	14	→	→	SYM
ejpam-3781	88	15	m2	m2	PROPN
ejpam-3781	88	16	be	be	VERB
ejpam-3781	88	17	a	a	DET
ejpam-3781	88	18	modified	modify	VERB
ejpam-3781	88	19	near	near	ADP
ejpam-3781	88	20	module	module	NOUN
ejpam-3781	88	21	homomorphism	homomorphism	NOUN
ejpam-3781	88	22	.	.	PUNCT
ejpam-3781	89	1	then	then	ADV
ejpam-3781	89	2	kerφ	kerφ	PROPN
ejpam-3781	89	3	is	be	AUX
ejpam-3781	89	4	an	an	DET
ejpam-3781	89	5	ideal	ideal	NOUN
ejpam-3781	89	6	of	of	ADP
ejpam-3781	89	7	m1	m1	PROPN
ejpam-3781	89	8	and	and	CCONJ
ejpam-3781	89	9	m1	m1	PROPN
ejpam-3781	89	10	kerφ	kerφ	PROPN
ejpam-3781	89	11	'	'	PART
ejpam-3781	89	12	φ(m1	φ(m1	NOUN
ejpam-3781	89	13	)	)	PUNCT
ejpam-3781	89	14	.	.	PUNCT
ejpam-3781	90	1	theorem	theorem	NOUN
ejpam-3781	90	2	3	3	NUM
ejpam-3781	90	3	.	.	PUNCT
ejpam-3781	91	1	the	the	DET
ejpam-3781	91	2	intersection	intersection	NOUN
ejpam-3781	91	3	of	of	ADP
ejpam-3781	91	4	any	any	DET
ejpam-3781	91	5	family	family	NOUN
ejpam-3781	91	6	of	of	ADP
ejpam-3781	91	7	ideals	ideal	NOUN
ejpam-3781	91	8	of	of	ADP
ejpam-3781	91	9	a	a	DET
ejpam-3781	91	10	modified	modify	VERB
ejpam-3781	91	11	near	near	ADP
ejpam-3781	91	12	module	module	NOUN
ejpam-3781	91	13	m	m	NOUN
ejpam-3781	91	14	is	be	AUX
ejpam-3781	91	15	ideal	ideal	ADJ
ejpam-3781	91	16	of	of	ADP
ejpam-3781	91	17	m	m	PROPN
ejpam-3781	91	18	.	.	PUNCT
ejpam-3781	92	1	proposition	proposition	NOUN
ejpam-3781	92	2	1	1	NUM
ejpam-3781	92	3	.	.	PUNCT
ejpam-3781	93	1	let	let	VERB
ejpam-3781	93	2	m	m	PRON
ejpam-3781	93	3	be	be	AUX
ejpam-3781	93	4	a	a	DET
ejpam-3781	93	5	modified	modify	VERB
ejpam-3781	93	6	near	near	ADP
ejpam-3781	93	7	module	module	NOUN
ejpam-3781	93	8	over	over	ADP
ejpam-3781	93	9	n	n	CCONJ
ejpam-3781	93	10	and	and	CCONJ
ejpam-3781	93	11	let	let	VERB
ejpam-3781	93	12	i	i	PRON
ejpam-3781	93	13	be	be	AUX
ejpam-3781	93	14	an	an	DET
ejpam-3781	93	15	ideal	ideal	NOUN
ejpam-3781	93	16	of	of	ADP
ejpam-3781	93	17	m	m	PROPN
ejpam-3781	93	18	.	.	PUNCT
ejpam-3781	94	1	let	let	VERB
ejpam-3781	94	2	m	m	PRON
ejpam-3781	94	3	i	i	ADJ
ejpam-3781	94	4	=	=	PUNCT
ejpam-3781	94	5	{	{	PUNCT
ejpam-3781	94	6	m+	m+	NUM
ejpam-3781	94	7	i|m	i|m	PROPN
ejpam-3781	94	8	∈m	∈m	NOUN
ejpam-3781	94	9	}	}	PUNCT
ejpam-3781	94	10	.	.	PUNCT
ejpam-3781	95	1	then	then	ADV
ejpam-3781	95	2	(	(	PUNCT
ejpam-3781	95	3	mi	mi	PROPN
ejpam-3781	95	4	,	,	PUNCT
ejpam-3781	95	5	⊕	⊕	PROPN
ejpam-3781	95	6	,	,	PUNCT
ejpam-3781	95	7	�	�	PROPN
ejpam-3781	95	8	)	)	PUNCT
ejpam-3781	95	9	is	be	AUX
ejpam-3781	95	10	a	a	DET
ejpam-3781	95	11	modified	modify	VERB
ejpam-3781	95	12	near	near	ADP
ejpam-3781	95	13	module	module	NOUN
ejpam-3781	95	14	when	when	SCONJ
ejpam-3781	95	15	⊕	⊕	PROPN
ejpam-3781	95	16	and	and	CCONJ
ejpam-3781	95	17	�	�	PROPN
ejpam-3781	95	18	are	be	AUX
ejpam-3781	95	19	defined	define	VERB
ejpam-3781	95	20	as	as	ADP
ejpam-3781	95	21	(	(	PUNCT
ejpam-3781	95	22	m+	m+	NUM
ejpam-3781	95	23	i)⊕	i)⊕	NOUN
ejpam-3781	95	24	(	(	PUNCT
ejpam-3781	95	25	m′	m′	NOUN
ejpam-3781	95	26	+	+	CCONJ
ejpam-3781	95	27	i	i	NOUN
ejpam-3781	95	28	)	)	PUNCT
ejpam-3781	96	1	=	=	SYM
ejpam-3781	96	2	(	(	PUNCT
ejpam-3781	96	3	m+m′	m+m′	ADJ
ejpam-3781	96	4	)	)	PUNCT
ejpam-3781	96	5	+	+	CCONJ
ejpam-3781	96	6	i	i	PRON
ejpam-3781	96	7	and	and	CCONJ
ejpam-3781	96	8	n	n	CCONJ
ejpam-3781	96	9	�	�	PROPN
ejpam-3781	96	10	(	(	PUNCT
ejpam-3781	96	11	m+	m+	NOUN
ejpam-3781	96	12	i	i	NOUN
ejpam-3781	96	13	)	)	PUNCT
ejpam-3781	96	14	=	=	PUNCT
ejpam-3781	97	1	nm+	nm+	NOUN
ejpam-3781	97	2	i	i	PRON
ejpam-3781	97	3	for	for	ADP
ejpam-3781	97	4	all	all	DET
ejpam-3781	97	5	m+	m+	NUM
ejpam-3781	97	6	i	i	PRON
ejpam-3781	97	7	,	,	PUNCT
ejpam-3781	97	8	m′	m′	PROPN
ejpam-3781	97	9	+	+	CCONJ
ejpam-3781	97	10	i	i	PRON
ejpam-3781	97	11	∈	∈	PROPN
ejpam-3781	97	12	m	m	VERB
ejpam-3781	97	13	i	i	NOUN
ejpam-3781	97	14	and	and	CCONJ
ejpam-3781	97	15	n	n	CCONJ
ejpam-3781	97	16	∈	∈	PROPN
ejpam-3781	97	17	n	n	NOUN
ejpam-3781	97	18	and	and	CCONJ
ejpam-3781	97	19	the	the	DET
ejpam-3781	97	20	natural	natural	ADJ
ejpam-3781	97	21	projection	projection	NOUN
ejpam-3781	97	22	map	map	NOUN
ejpam-3781	97	23	π	π	X
ejpam-3781	97	24	:	:	PUNCT
ejpam-3781	97	25	m	m	VERB
ejpam-3781	97	26	→	→	NOUN
ejpam-3781	97	27	m	m	AUX
ejpam-3781	97	28	i	i	PRON
ejpam-3781	97	29	defined	define	VERB
ejpam-3781	97	30	by	by	ADP
ejpam-3781	97	31	π(m	π(m	NOUN
ejpam-3781	97	32	)	)	PUNCT
ejpam-3781	97	33	=	=	PRON
ejpam-3781	98	1	m	m	VERB
ejpam-3781	98	2	+	+	ADJ
ejpam-3781	98	3	i	i	PRON
ejpam-3781	98	4	is	be	AUX
ejpam-3781	98	5	a	a	DET
ejpam-3781	98	6	modified	modify	VERB
ejpam-3781	98	7	near	near	ADP
ejpam-3781	98	8	module	module	NOUN
ejpam-3781	98	9	homomorphism	homomorphism	NOUN
ejpam-3781	98	10	with	with	ADP
ejpam-3781	98	11	kernel	kernel	PROPN
ejpam-3781	98	12	i.	i.	PROPN
ejpam-3781	98	13	3	3	PROPN
ejpam-3781	98	14	.	.	PUNCT
ejpam-3781	99	1	near	near	ADP
ejpam-3781	99	2	ring	ring	NOUN
ejpam-3781	99	3	multiplication	multiplication	NOUN
ejpam-3781	99	4	on	on	ADP
ejpam-3781	99	5	a	a	DET
ejpam-3781	99	6	modified	modify	VERB
ejpam-3781	99	7	near	near	ADP
ejpam-3781	99	8	module	module	NOUN
ejpam-3781	99	9	the	the	DET
ejpam-3781	99	10	following	follow	VERB
ejpam-3781	99	11	theorem	theorem	NOUN
ejpam-3781	99	12	explains	explain	VERB
ejpam-3781	99	13	a	a	DET
ejpam-3781	99	14	method	method	NOUN
ejpam-3781	99	15	of	of	ADP
ejpam-3781	99	16	obtaining	obtain	VERB
ejpam-3781	99	17	a	a	DET
ejpam-3781	99	18	near	near	ADJ
ejpam-3781	99	19	ring	ring	NOUN
ejpam-3781	99	20	multiplications	multiplication	NOUN
ejpam-3781	99	21	on	on	ADP
ejpam-3781	99	22	a	a	DET
ejpam-3781	99	23	modified	modify	VERB
ejpam-3781	99	24	near	near	ADP
ejpam-3781	99	25	module	module	NOUN
ejpam-3781	99	26	over	over	ADP
ejpam-3781	99	27	n	n	NOUN
ejpam-3781	99	28	via	via	ADP
ejpam-3781	99	29	semilinear	semilinear	NOUN
ejpam-3781	99	30	map	map	NOUN
ejpam-3781	99	31	from	from	ADP
ejpam-3781	99	32	m	m	PROPN
ejpam-3781	99	33	into	into	ADP
ejpam-3781	99	34	n	n	PROPN
ejpam-3781	99	35	.	.	PUNCT
ejpam-3781	100	1	definition	definition	NOUN
ejpam-3781	100	2	6	6	NUM
ejpam-3781	100	3	.	.	PUNCT
ejpam-3781	101	1	let	let	AUX
ejpam-3781	101	2	(	(	PUNCT
ejpam-3781	101	3	m,+	m,+	INTJ
ejpam-3781	101	4	,	,	PUNCT
ejpam-3781	101	5	·	·	PUNCT
ejpam-3781	101	6	)	)	PUNCT
ejpam-3781	101	7	be	be	AUX
ejpam-3781	101	8	a	a	DET
ejpam-3781	101	9	modified	modify	VERB
ejpam-3781	101	10	near	near	ADP
ejpam-3781	101	11	module	module	NOUN
ejpam-3781	101	12	over	over	ADP
ejpam-3781	101	13	n	n	PROPN
ejpam-3781	101	14	.	.	PUNCT
ejpam-3781	102	1	we	we	PRON
ejpam-3781	102	2	call	call	VERB
ejpam-3781	102	3	a	a	DET
ejpam-3781	102	4	mapping	mapping	NOUN
ejpam-3781	102	5	f	f	NOUN
ejpam-3781	102	6	:	:	PUNCT
ejpam-3781	102	7	m	m	VERB
ejpam-3781	102	8	→	→	SYM
ejpam-3781	102	9	n	n	PROPN
ejpam-3781	102	10	a	a	DET
ejpam-3781	102	11	semilinear	semilinear	NOUN
ejpam-3781	102	12	if	if	SCONJ
ejpam-3781	102	13	f(f(m1)m2	f(f(m1)m2	ADJ
ejpam-3781	102	14	)	)	PUNCT
ejpam-3781	102	15	=	=	SYM
ejpam-3781	102	16	f(m1)f(m2	f(m1)f(m2	PROPN
ejpam-3781	102	17	)	)	PUNCT
ejpam-3781	102	18	for	for	ADP
ejpam-3781	102	19	all	all	DET
ejpam-3781	102	20	m1,m2	m1,m2	PROPN
ejpam-3781	102	21	∈m	∈m	NOUN
ejpam-3781	102	22	.	.	PUNCT
ejpam-3781	103	1	theorem	theorem	ADJ
ejpam-3781	103	2	4	4	NUM
ejpam-3781	103	3	.	.	PUNCT
ejpam-3781	104	1	let	let	AUX
ejpam-3781	104	2	(	(	PUNCT
ejpam-3781	104	3	m,+	m,+	INTJ
ejpam-3781	104	4	,	,	PUNCT
ejpam-3781	104	5	·	·	PUNCT
ejpam-3781	104	6	)	)	PUNCT
ejpam-3781	104	7	be	be	AUX
ejpam-3781	104	8	a	a	DET
ejpam-3781	104	9	modified	modify	VERB
ejpam-3781	104	10	near	near	ADP
ejpam-3781	104	11	module	module	NOUN
ejpam-3781	104	12	over	over	ADP
ejpam-3781	104	13	a	a	DET
ejpam-3781	104	14	near	near	ADJ
ejpam-3781	104	15	ring	ring	NOUN
ejpam-3781	104	16	(	(	PUNCT
ejpam-3781	104	17	n,+	n,+	NUM
ejpam-3781	104	18	,	,	PUNCT
ejpam-3781	104	19	·	·	PUNCT
ejpam-3781	104	20	)	)	PUNCT
ejpam-3781	104	21	.	.	PUNCT
ejpam-3781	105	1	let	let	VERB
ejpam-3781	105	2	f	f	PRON
ejpam-3781	105	3	be	be	AUX
ejpam-3781	105	4	a	a	DET
ejpam-3781	105	5	semilinear	semilinear	ADJ
ejpam-3781	105	6	map	map	NOUN
ejpam-3781	105	7	from	from	ADP
ejpam-3781	105	8	m	m	PROPN
ejpam-3781	105	9	into	into	ADP
ejpam-3781	105	10	n	n	PROPN
ejpam-3781	105	11	.	.	PUNCT
ejpam-3781	106	1	define	define	VERB
ejpam-3781	106	2	the	the	DET
ejpam-3781	106	3	binary	binary	PROPN
ejpam-3781	106	4	operation	operation	NOUN
ejpam-3781	106	5	∗	∗	NOUN
ejpam-3781	106	6	on	on	ADP
ejpam-3781	106	7	m	m	NOUN
ejpam-3781	106	8	as	as	ADP
ejpam-3781	106	9	m1∗m2	m1∗m2	PROPN
ejpam-3781	106	10	=	=	SYM
ejpam-3781	106	11	f(m2)m1	f(m2)m1	PROPN
ejpam-3781	106	12	for	for	ADP
ejpam-3781	106	13	all	all	DET
ejpam-3781	106	14	m1,m2	m1,m2	PROPN
ejpam-3781	106	15	∈m	∈m	NOUN
ejpam-3781	106	16	.	.	PUNCT
ejpam-3781	107	1	then	then	ADV
ejpam-3781	107	2	(	(	PUNCT
ejpam-3781	107	3	m,+	m,+	INTJ
ejpam-3781	107	4	,	,	PUNCT
ejpam-3781	107	5	∗	∗	NOUN
ejpam-3781	107	6	)	)	PUNCT
ejpam-3781	107	7	is	be	AUX
ejpam-3781	107	8	a	a	DET
ejpam-3781	107	9	near	near	ADJ
ejpam-3781	107	10	ring	ring	NOUN
ejpam-3781	107	11	.	.	PUNCT
ejpam-3781	108	1	proof	proof	NOUN
ejpam-3781	108	2	.	.	PUNCT
ejpam-3781	109	1	for	for	ADP
ejpam-3781	109	2	any	any	DET
ejpam-3781	109	3	m1,m2,m3	m1,m2,m3	ADJ
ejpam-3781	109	4	∈m	∈m	NOUN
ejpam-3781	109	5	,	,	PUNCT
ejpam-3781	109	6	m1	m1	PROPN
ejpam-3781	109	7	∗	∗	NOUN
ejpam-3781	109	8	(	(	PUNCT
ejpam-3781	109	9	m2	m2	PROPN
ejpam-3781	109	10	∗m3	∗m3	PROPN
ejpam-3781	109	11	)	)	PUNCT
ejpam-3781	109	12	=	=	PUNCT
ejpam-3781	109	13	f(m2	f(m2	NOUN
ejpam-3781	109	14	∗m3)m1	∗m3)m1	NOUN
ejpam-3781	109	15	=	=	SYM
ejpam-3781	109	16	f(f(m3)m2)m1	f(f(m3)m2)m1	NOUN
ejpam-3781	109	17	=	=	SYM
ejpam-3781	110	1	[	[	X
ejpam-3781	110	2	f(m3)f(m2)]m1	f(m3)f(m2)]m1	NOUN
ejpam-3781	110	3	and	and	CCONJ
ejpam-3781	110	4	(	(	PUNCT
ejpam-3781	110	5	m1	m1	PROPN
ejpam-3781	110	6	∗m2	∗m2	PROPN
ejpam-3781	110	7	)	)	PUNCT
ejpam-3781	110	8	∗m3	∗m3	PROPN
ejpam-3781	110	9	=	=	SYM
ejpam-3781	110	10	f(m3)(m1	f(m3)(m1	NOUN
ejpam-3781	110	11	∗m2	∗m2	PROPN
ejpam-3781	110	12	)	)	PUNCT
ejpam-3781	110	13	=	=	SYM
ejpam-3781	110	14	f(m3)[f(m2)m1	f(m3)[f(m2)m1	NOUN
ejpam-3781	110	15	]	]	PUNCT
ejpam-3781	110	16	=	=	PUNCT
ejpam-3781	111	1	[	[	X
ejpam-3781	111	2	f(m3)f(m2)]m1	f(m3)f(m2)]m1	NOUN
ejpam-3781	111	3	.	.	PUNCT
ejpam-3781	112	1	so	so	ADV
ejpam-3781	112	2	the	the	DET
ejpam-3781	112	3	binary	binary	PROPN
ejpam-3781	112	4	operation	operation	NOUN
ejpam-3781	112	5	∗	∗	NOUN
ejpam-3781	112	6	is	be	AUX
ejpam-3781	112	7	associative	associative	ADJ
ejpam-3781	112	8	.	.	PUNCT
ejpam-3781	113	1	now	now	ADV
ejpam-3781	113	2	(	(	PUNCT
ejpam-3781	113	3	m1	m1	PROPN
ejpam-3781	113	4	+	+	CCONJ
ejpam-3781	113	5	m2	m2	PROPN
ejpam-3781	113	6	)	)	PUNCT
ejpam-3781	113	7	∗m3	∗m3	PROPN
ejpam-3781	113	8	=	=	SYM
ejpam-3781	113	9	f(m3)(m1	f(m3)(m1	PROPN
ejpam-3781	113	10	+	+	NUM
ejpam-3781	113	11	m2	m2	PROPN
ejpam-3781	113	12	)	)	PUNCT
ejpam-3781	113	13	=	=	PUNCT
ejpam-3781	113	14	f(m3)m1	f(m3)m1	NOUN
ejpam-3781	113	15	+	+	CCONJ
ejpam-3781	113	16	f(m3)m2	f(m3)m2	NOUN
ejpam-3781	113	17	=	=	SYM
ejpam-3781	113	18	m1	m1	PROPN
ejpam-3781	113	19	∗m3	∗m3	PROPN
ejpam-3781	113	20	+	+	CCONJ
ejpam-3781	113	21	m2	m2	PROPN
ejpam-3781	113	22	∗m3	∗m3	PROPN
ejpam-3781	113	23	.	.	PUNCT
ejpam-3781	114	1	so	so	ADV
ejpam-3781	114	2	the	the	DET
ejpam-3781	114	3	binary	binary	PROPN
ejpam-3781	114	4	operation	operation	NOUN
ejpam-3781	114	5	∗	∗	NOUN
ejpam-3781	114	6	is	be	AUX
ejpam-3781	114	7	right	right	ADV
ejpam-3781	114	8	distributive	distributive	ADJ
ejpam-3781	114	9	and	and	CCONJ
ejpam-3781	114	10	hence	hence	ADV
ejpam-3781	114	11	(	(	PUNCT
ejpam-3781	114	12	m,+	m,+	INTJ
ejpam-3781	114	13	,	,	PUNCT
ejpam-3781	114	14	∗	∗	NOUN
ejpam-3781	114	15	)	)	PUNCT
ejpam-3781	114	16	is	be	AUX
ejpam-3781	114	17	a	a	DET
ejpam-3781	114	18	near	near	ADJ
ejpam-3781	114	19	ring	ring	NOUN
ejpam-3781	114	20	.	.	PUNCT
ejpam-3781	115	1	examples	example	NOUN
ejpam-3781	115	2	3.3	3.3	NUM
ejpam-3781	115	3	through	through	ADP
ejpam-3781	115	4	3.7	3.7	NUM
ejpam-3781	115	5	illustrate	illustrate	VERB
ejpam-3781	115	6	the	the	DET
ejpam-3781	115	7	technique	technique	NOUN
ejpam-3781	115	8	of	of	ADP
ejpam-3781	115	9	defining	define	VERB
ejpam-3781	115	10	a	a	DET
ejpam-3781	115	11	near	near	ADJ
ejpam-3781	115	12	ring	ring	NOUN
ejpam-3781	115	13	multiplication	multiplication	NOUN
ejpam-3781	115	14	on	on	ADP
ejpam-3781	115	15	(	(	PUNCT
ejpam-3781	115	16	m,+	m,+	NOUN
ejpam-3781	115	17	)	)	PUNCT
ejpam-3781	115	18	example	example	NOUN
ejpam-3781	116	1	4	4	NUM
ejpam-3781	116	2	.	.	PUNCT
ejpam-3781	117	1	let	let	VERB
ejpam-3781	117	2	m	m	VERB
ejpam-3781	117	3	=	=	VERB
ejpam-3781	117	4	{	{	PUNCT
ejpam-3781	117	5	f	f	NOUN
ejpam-3781	117	6	|f	|f	PROPN
ejpam-3781	117	7	:	:	PUNCT
ejpam-3781	117	8	r→	r→	PROPN
ejpam-3781	117	9	r	r	NOUN
ejpam-3781	117	10	}	}	PUNCT
ejpam-3781	117	11	and	and	CCONJ
ejpam-3781	117	12	n	n	NOUN
ejpam-3781	117	13	=	=	SYM
ejpam-3781	117	14	(	(	PUNCT
ejpam-3781	117	15	end(r,+),+	end(r,+),+	NOUN
ejpam-3781	117	16	,	,	PUNCT
ejpam-3781	117	17	◦	◦	NOUN
ejpam-3781	117	18	)	)	PUNCT
ejpam-3781	117	19	.	.	PUNCT
ejpam-3781	118	1	then	then	ADV
ejpam-3781	118	2	(	(	PUNCT
ejpam-3781	118	3	m,+	m,+	INTJ
ejpam-3781	118	4	,	,	PUNCT
ejpam-3781	118	5	◦	◦	NOUN
ejpam-3781	118	6	)	)	PUNCT
ejpam-3781	118	7	is	be	AUX
ejpam-3781	118	8	a	a	DET
ejpam-3781	118	9	modified	modify	VERB
ejpam-3781	118	10	near	near	ADP
ejpam-3781	118	11	module	module	NOUN
ejpam-3781	118	12	over	over	ADP
ejpam-3781	118	13	(	(	PUNCT
ejpam-3781	118	14	n,+	n,+	NUM
ejpam-3781	118	15	,	,	PUNCT
ejpam-3781	118	16	◦	◦	NOUN
ejpam-3781	118	17	)	)	PUNCT
ejpam-3781	118	18	.	.	PUNCT
ejpam-3781	119	1	define	define	VERB
ejpam-3781	119	2	α	α	NOUN
ejpam-3781	119	3	:	:	PUNCT
ejpam-3781	119	4	m	m	VERB
ejpam-3781	119	5	→	→	SYM
ejpam-3781	119	6	n	n	X
ejpam-3781	119	7	by	by	ADP
ejpam-3781	119	8	α(f	α(f	PROPN
ejpam-3781	119	9	)	)	PUNCT
ejpam-3781	119	10	=	=	PUNCT
ejpam-3781	120	1	f	f	X
ejpam-3781	121	1	′	′	INTJ
ejpam-3781	121	2	where	where	SCONJ
ejpam-3781	121	3	f	f	PROPN
ejpam-3781	121	4	′(x	′(x	NOUN
ejpam-3781	121	5	)	)	PUNCT
ejpam-3781	121	6	=	=	PRON
ejpam-3781	121	7	{	{	PUNCT
ejpam-3781	121	8	0	0	NUM
ejpam-3781	121	9	if	if	SCONJ
ejpam-3781	121	10	x	x	X
ejpam-3781	121	11	=	=	SYM
ejpam-3781	121	12	0	0	NUM
ejpam-3781	121	13	f(x	f(x	PROPN
ejpam-3781	121	14	)	)	PUNCT
ejpam-3781	122	1	if	if	SCONJ
ejpam-3781	122	2	x	x	PROPN
ejpam-3781	122	3	6=	6=	ADP
ejpam-3781	122	4	0	0	NUM
ejpam-3781	122	5	.	.	PUNCT
ejpam-3781	122	6	note	note	VERB
ejpam-3781	122	7	that	that	SCONJ
ejpam-3781	122	8	f(x	f(x	NOUN
ejpam-3781	122	9	)	)	PUNCT
ejpam-3781	123	1	=	=	SYM
ejpam-3781	123	2	0	0	NUM
ejpam-3781	123	3	implies	imply	VERB
ejpam-3781	123	4	f	f	PROPN
ejpam-3781	123	5	′(x	′(x	PROPN
ejpam-3781	123	6	)	)	PUNCT
ejpam-3781	123	7	=	=	SYM
ejpam-3781	123	8	0	0	NUM
ejpam-3781	123	9	for	for	SCONJ
ejpam-3781	123	10	x	x	PROPN
ejpam-3781	123	11	∈	∈	PROPN
ejpam-3781	123	12	r.	r.	NOUN
ejpam-3781	123	13	we	we	PRON
ejpam-3781	123	14	claim	claim	VERB
ejpam-3781	123	15	that	that	SCONJ
ejpam-3781	123	16	α	α	PROPN
ejpam-3781	123	17	is	be	AUX
ejpam-3781	123	18	semilinear	semilinear	NOUN
ejpam-3781	123	19	.	.	PUNCT
ejpam-3781	124	1	a.v	a.v	PROPN
ejpam-3781	124	2	.	.	PROPN
ejpam-3781	124	3	ramakrishna	ramakrishna	PROPN
ejpam-3781	124	4	,	,	PUNCT
ejpam-3781	124	5	t.v.n	t.v.n	PROPN
ejpam-3781	124	6	.	.	PUNCT
ejpam-3781	125	1	prasanna	prasanna	PROPN
ejpam-3781	125	2	,	,	PUNCT
ejpam-3781	125	3	d.v	d.v	PROPN
ejpam-3781	125	4	.	.	PROPN
ejpam-3781	125	5	lakshmi	lakshmi	PROPN
ejpam-3781	125	6	/	/	SYM
ejpam-3781	125	7	eur	eur	PROPN
ejpam-3781	125	8	.	.	PUNCT
ejpam-3781	126	1	j.	j.	PROPN
ejpam-3781	126	2	pure	pure	PROPN
ejpam-3781	126	3	appl	appl	PROPN
ejpam-3781	126	4	.	.	PROPN
ejpam-3781	126	5	math	math	PROPN
ejpam-3781	126	6	,	,	PUNCT
ejpam-3781	126	7	14	14	NUM
ejpam-3781	126	8	(	(	PUNCT
ejpam-3781	126	9	1	1	NUM
ejpam-3781	126	10	)	)	PUNCT
ejpam-3781	126	11	(	(	PUNCT
ejpam-3781	126	12	2021	2021	NUM
ejpam-3781	126	13	)	)	PUNCT
ejpam-3781	126	14	,	,	PUNCT
ejpam-3781	126	15	126	126	NUM
ejpam-3781	126	16	-	-	SYM
ejpam-3781	126	17	134	134	NUM
ejpam-3781	126	18	130	130	NUM
ejpam-3781	126	19	let	let	VERB
ejpam-3781	126	20	f	f	NOUN
ejpam-3781	126	21	,	,	PUNCT
ejpam-3781	126	22	g	g	NOUN
ejpam-3781	126	23	∈m	∈m	NOUN
ejpam-3781	126	24	and	and	CCONJ
ejpam-3781	126	25	x	x	PROPN
ejpam-3781	126	26	∈	∈	PROPN
ejpam-3781	126	27	r.	r.	PROPN
ejpam-3781	126	28	case(i	case(i	PROPN
ejpam-3781	126	29	):	):	PUNCT
ejpam-3781	126	30	x	x	PROPN
ejpam-3781	126	31	6=	6=	ADP
ejpam-3781	126	32	0	0	NUM
ejpam-3781	126	33	.	.	PUNCT
ejpam-3781	127	1	now	now	ADV
ejpam-3781	127	2	α(α(f)og)(x	α(α(f)og)(x	NUM
ejpam-3781	127	3	)	)	PUNCT
ejpam-3781	127	4	=	=	SYM
ejpam-3781	127	5	α(f	α(f	PROPN
ejpam-3781	127	6	′og)(x	′og)(x	PROPN
ejpam-3781	127	7	)	)	PUNCT
ejpam-3781	128	1	=	=	PRON
ejpam-3781	128	2	(	(	PUNCT
ejpam-3781	128	3	f	f	NOUN
ejpam-3781	128	4	′	′	NUM
ejpam-3781	128	5	◦	◦	NOUN
ejpam-3781	128	6	g)′(x	g)′(x	NOUN
ejpam-3781	128	7	)	)	PUNCT
ejpam-3781	128	8	=	=	PUNCT
ejpam-3781	129	1	(	(	PUNCT
ejpam-3781	129	2	f	f	NOUN
ejpam-3781	129	3	′	′	NUM
ejpam-3781	129	4	◦	◦	PROPN
ejpam-3781	129	5	g)(x	g)(x	PROPN
ejpam-3781	129	6	)	)	PUNCT
ejpam-3781	129	7	=	=	SYM
ejpam-3781	129	8	f	f	NOUN
ejpam-3781	129	9	′(g(x	′(g(x	NOUN
ejpam-3781	129	10	)	)	PUNCT
ejpam-3781	129	11	)	)	PUNCT
ejpam-3781	130	1	=	=	SYM
ejpam-3781	130	2	f	f	X
ejpam-3781	130	3	′(g′(x	′(g′(x	PROPN
ejpam-3781	130	4	)	)	PUNCT
ejpam-3781	130	5	)	)	PUNCT
ejpam-3781	131	1	=	=	PUNCT
ejpam-3781	131	2	(	(	PUNCT
ejpam-3781	131	3	f	f	NOUN
ejpam-3781	131	4	′	′	NUM
ejpam-3781	131	5	◦	◦	NOUN
ejpam-3781	131	6	g′)(x	g′)(x	NOUN
ejpam-3781	131	7	)	)	PUNCT
ejpam-3781	132	1	=	=	SYM
ejpam-3781	133	1	(	(	PUNCT
ejpam-3781	133	2	α(f	α(f	PROPN
ejpam-3781	133	3	)	)	PUNCT
ejpam-3781	133	4	◦	◦	PROPN
ejpam-3781	133	5	α(g))(x	α(g))(x	NOUN
ejpam-3781	133	6	)	)	PUNCT
ejpam-3781	133	7	implies	imply	VERB
ejpam-3781	133	8	α(α(f	α(α(f	PROPN
ejpam-3781	133	9	)	)	PUNCT
ejpam-3781	133	10	◦	◦	NOUN
ejpam-3781	133	11	g	g	NOUN
ejpam-3781	133	12	)	)	PUNCT
ejpam-3781	133	13	=	=	SYM
ejpam-3781	133	14	α(f	α(f	PROPN
ejpam-3781	133	15	)	)	PUNCT
ejpam-3781	133	16	◦	◦	NOUN
ejpam-3781	133	17	α(g	α(g	NUM
ejpam-3781	133	18	)	)	PUNCT
ejpam-3781	133	19	.	.	PUNCT
ejpam-3781	134	1	case(ii	case(ii	VERB
ejpam-3781	134	2	):	):	PUNCT
ejpam-3781	134	3	x	x	SYM
ejpam-3781	134	4	=	=	SYM
ejpam-3781	134	5	0	0	PROPN
ejpam-3781	134	6	.	.	PUNCT
ejpam-3781	135	1	now	now	ADV
ejpam-3781	135	2	α(α(f	α(α(f	PROPN
ejpam-3781	135	3	)	)	PUNCT
ejpam-3781	135	4	◦	◦	NOUN
ejpam-3781	135	5	g)(0	g)(0	NUM
ejpam-3781	135	6	)	)	PUNCT
ejpam-3781	136	1	=	=	SYM
ejpam-3781	136	2	α(f	α(f	NOUN
ejpam-3781	136	3	′	′	NUM
ejpam-3781	136	4	◦	◦	NOUN
ejpam-3781	136	5	g)(0	g)(0	NUM
ejpam-3781	136	6	)	)	PUNCT
ejpam-3781	137	1	=	=	SYM
ejpam-3781	137	2	(	(	PUNCT
ejpam-3781	137	3	f	f	NOUN
ejpam-3781	137	4	′	′	NUM
ejpam-3781	137	5	◦	◦	NOUN
ejpam-3781	137	6	g)′(0	g)′(0	NOUN
ejpam-3781	137	7	)	)	PUNCT
ejpam-3781	137	8	=	=	SYM
ejpam-3781	137	9	0	0	X
ejpam-3781	137	10	.	.	PUNCT
ejpam-3781	138	1	also	also	ADV
ejpam-3781	138	2	[	[	X
ejpam-3781	138	3	α(f	α(f	PROPN
ejpam-3781	138	4	)	)	PUNCT
ejpam-3781	138	5	◦	◦	NOUN
ejpam-3781	138	6	α(g)](0	α(g)](0	NUM
ejpam-3781	138	7	)	)	PUNCT
ejpam-3781	139	1	=	=	PUNCT
ejpam-3781	139	2	(	(	PUNCT
ejpam-3781	139	3	f	f	NOUN
ejpam-3781	139	4	′	′	NUM
ejpam-3781	139	5	◦	◦	NOUN
ejpam-3781	139	6	g′)(0	g′)(0	NOUN
ejpam-3781	139	7	)	)	PUNCT
ejpam-3781	139	8	=	=	SYM
ejpam-3781	139	9	f	f	X
ejpam-3781	139	10	′(g′(0	′(g′(0	NOUN
ejpam-3781	139	11	)	)	PUNCT
ejpam-3781	139	12	)	)	PUNCT
ejpam-3781	140	1	=	=	PUNCT
ejpam-3781	140	2	f	f	PROPN
ejpam-3781	140	3	′(0	′(0	PROPN
ejpam-3781	140	4	)	)	PUNCT
ejpam-3781	141	1	=	=	SYM
ejpam-3781	141	2	0	0	X
ejpam-3781	141	3	.	.	PUNCT
ejpam-3781	142	1	so	so	ADV
ejpam-3781	142	2	α(α(f	α(α(f	PROPN
ejpam-3781	142	3	)	)	PUNCT
ejpam-3781	142	4	◦	◦	NOUN
ejpam-3781	142	5	g	g	NOUN
ejpam-3781	142	6	)	)	PUNCT
ejpam-3781	142	7	=	=	SYM
ejpam-3781	142	8	α(f	α(f	PROPN
ejpam-3781	142	9	)	)	PUNCT
ejpam-3781	142	10	◦	◦	NOUN
ejpam-3781	142	11	α(g	α(g	NUM
ejpam-3781	142	12	)	)	PUNCT
ejpam-3781	143	1	when	when	SCONJ
ejpam-3781	143	2	x	x	X
ejpam-3781	144	1	=	=	NOUN
ejpam-3781	145	1	0	0	X
ejpam-3781	145	2	.	.	PUNCT
ejpam-3781	146	1	hence	hence	ADV
ejpam-3781	146	2	α	α	PROPN
ejpam-3781	146	3	is	be	AUX
ejpam-3781	146	4	semilinear	semilinear	ADJ
ejpam-3781	146	5	;	;	PUNCT
ejpam-3781	146	6	therefore	therefore	ADV
ejpam-3781	146	7	(	(	PUNCT
ejpam-3781	146	8	m,+	m,+	INTJ
ejpam-3781	146	9	,	,	PUNCT
ejpam-3781	146	10	∗	∗	NOUN
ejpam-3781	146	11	)	)	PUNCT
ejpam-3781	146	12	is	be	AUX
ejpam-3781	146	13	a	a	DET
ejpam-3781	146	14	near	near	ADJ
ejpam-3781	146	15	ring	ring	NOUN
ejpam-3781	146	16	with	with	ADP
ejpam-3781	146	17	∗	∗	NOUN
ejpam-3781	146	18	defined	define	VERB
ejpam-3781	146	19	by	by	ADP
ejpam-3781	146	20	f	f	PROPN
ejpam-3781	146	21	∗g	∗g	PROPN
ejpam-3781	146	22	=	=	PROPN
ejpam-3781	147	1	α(g)	α(g)	PROPN
ejpam-3781	147	2	◦	◦	NOUN
ejpam-3781	147	3	f	f	NOUN
ejpam-3781	147	4	=	=	SYM
ejpam-3781	147	5	g′	g′	PROPN
ejpam-3781	148	1	◦	◦	NOUN
ejpam-3781	148	2	f	f	X
ejpam-3781	148	3	.	.	PUNCT
ejpam-3781	149	1	example	example	NOUN
ejpam-3781	150	1	5	5	NUM
ejpam-3781	150	2	.	.	PUNCT
ejpam-3781	151	1	let	let	VERB
ejpam-3781	151	2	m	m	PRON
ejpam-3781	151	3	be	be	AUX
ejpam-3781	151	4	the	the	DET
ejpam-3781	151	5	abelian	abelian	ADJ
ejpam-3781	151	6	group	group	NOUN
ejpam-3781	151	7	of	of	ADP
ejpam-3781	151	8	all	all	DET
ejpam-3781	151	9	n×	n×	PROPN
ejpam-3781	151	10	n	n	PRON
ejpam-3781	151	11	circulant	circulant	ADJ
ejpam-3781	151	12	matrices	matrix	NOUN
ejpam-3781	151	13	with	with	ADP
ejpam-3781	151	14	real	real	ADJ
ejpam-3781	151	15	entries	entry	NOUN
ejpam-3781	151	16	.	.	PUNCT
ejpam-3781	152	1	then	then	ADV
ejpam-3781	152	2	(	(	PUNCT
ejpam-3781	152	3	m,+	m,+	X
ejpam-3781	152	4	)	)	PUNCT
ejpam-3781	152	5	is	be	AUX
ejpam-3781	152	6	a	a	DET
ejpam-3781	152	7	modified	modify	VERB
ejpam-3781	152	8	near	near	ADP
ejpam-3781	152	9	module	module	NOUN
ejpam-3781	152	10	over	over	ADP
ejpam-3781	152	11	n	n	NOUN
ejpam-3781	152	12	=	=	PUNCT
ejpam-3781	152	13	(	(	PUNCT
ejpam-3781	152	14	r,+	r,+	NUM
ejpam-3781	152	15	,	,	PUNCT
ejpam-3781	152	16	·	·	PUNCT
ejpam-3781	152	17	)	)	PUNCT
ejpam-3781	152	18	,	,	PUNCT
ejpam-3781	152	19	if	if	SCONJ
ejpam-3781	152	20	we	we	PRON
ejpam-3781	152	21	define	define	VERB
ejpam-3781	152	22	ka	ka	PROPN
ejpam-3781	152	23	as	as	ADP
ejpam-3781	152	24	the	the	DET
ejpam-3781	152	25	matrix	matrix	NOUN
ejpam-3781	152	26	obtained	obtain	VERB
ejpam-3781	152	27	by	by	ADP
ejpam-3781	152	28	multiplying	multiply	VERB
ejpam-3781	152	29	each	each	DET
ejpam-3781	152	30	entry	entry	NOUN
ejpam-3781	152	31	of	of	ADP
ejpam-3781	152	32	a	a	PRON
ejpam-3781	152	33	by	by	ADP
ejpam-3781	152	34	k.	k.	NOUN
ejpam-3781	152	35	define	define	VERB
ejpam-3781	152	36	α	α	NOUN
ejpam-3781	152	37	:	:	PUNCT
ejpam-3781	152	38	m	m	VERB
ejpam-3781	152	39	→	→	SYM
ejpam-3781	152	40	n	n	X
ejpam-3781	152	41	by	by	ADP
ejpam-3781	152	42	α(a	α(a	NOUN
ejpam-3781	152	43	)	)	PUNCT
ejpam-3781	153	1	=	=	VERB
ejpam-3781	153	2	speca	speca	VERB
ejpam-3781	153	3	for	for	ADP
ejpam-3781	153	4	all	all	DET
ejpam-3781	153	5	a	a	DET
ejpam-3781	153	6	∈m	∈m	NOUN
ejpam-3781	153	7	where	where	SCONJ
ejpam-3781	153	8	speca	speca	NOUN
ejpam-3781	153	9	=	=	PRON
ejpam-3781	153	10	max{|λi||λi	max{|λi||λi	PROPN
ejpam-3781	153	11	is	be	AUX
ejpam-3781	153	12	an	an	DET
ejpam-3781	153	13	eigen	eigen	NOUN
ejpam-3781	153	14	value	value	NOUN
ejpam-3781	153	15	of	of	ADP
ejpam-3781	153	16	a	a	PRON
ejpam-3781	153	17	}	}	PUNCT
ejpam-3781	153	18	.	.	PUNCT
ejpam-3781	154	1	now	now	ADV
ejpam-3781	154	2	α(α(a)b	α(α(a)b	NOUN
ejpam-3781	154	3	)	)	PUNCT
ejpam-3781	155	1	=	=	SYM
ejpam-3781	155	2	spec(α(a)b	spec(α(a)b	X
ejpam-3781	155	3	)	)	PUNCT
ejpam-3781	155	4	=	=	SYM
ejpam-3781	155	5	α(a)specb	α(a)specb	NOUN
ejpam-3781	155	6	=	=	SYM
ejpam-3781	155	7	α(a)α(b	α(a)α(b	NOUN
ejpam-3781	155	8	)	)	PUNCT
ejpam-3781	155	9	for	for	ADP
ejpam-3781	155	10	all	all	DET
ejpam-3781	155	11	a	a	DET
ejpam-3781	155	12	,	,	PUNCT
ejpam-3781	155	13	b	b	NOUN
ejpam-3781	155	14	∈m	∈m	NOUN
ejpam-3781	155	15	.	.	PUNCT
ejpam-3781	156	1	hence	hence	ADV
ejpam-3781	156	2	α	α	PROPN
ejpam-3781	156	3	is	be	AUX
ejpam-3781	156	4	semilinear	semilinear	ADJ
ejpam-3781	156	5	;	;	PUNCT
ejpam-3781	156	6	therefore	therefore	ADV
ejpam-3781	156	7	(	(	PUNCT
ejpam-3781	156	8	m,+	m,+	INTJ
ejpam-3781	156	9	,	,	PUNCT
ejpam-3781	156	10	∗	∗	NOUN
ejpam-3781	156	11	)	)	PUNCT
ejpam-3781	156	12	is	be	AUX
ejpam-3781	156	13	a	a	DET
ejpam-3781	156	14	near	near	ADJ
ejpam-3781	156	15	ring	ring	NOUN
ejpam-3781	156	16	with	with	ADP
ejpam-3781	156	17	∗	∗	NOUN
ejpam-3781	156	18	defined	define	VERB
ejpam-3781	156	19	by	by	ADP
ejpam-3781	156	20	a∗b	a∗b	PROPN
ejpam-3781	156	21	=	=	SYM
ejpam-3781	156	22	α(b)a	α(b)a	PROPN
ejpam-3781	156	23	=	=	SYM
ejpam-3781	156	24	specb	specb	NOUN
ejpam-3781	156	25	a.	a.	NOUN
ejpam-3781	156	26	example	example	NOUN
ejpam-3781	156	27	6	6	NUM
ejpam-3781	156	28	.	.	PUNCT
ejpam-3781	157	1	let	let	AUX
ejpam-3781	157	2	(	(	PUNCT
ejpam-3781	157	3	g,+	g,+	PROPN
ejpam-3781	157	4	)	)	PUNCT
ejpam-3781	157	5	be	be	AUX
ejpam-3781	157	6	a	a	DET
ejpam-3781	157	7	(	(	PUNCT
ejpam-3781	157	8	not	not	PART
ejpam-3781	157	9	necessarily	necessarily	ADV
ejpam-3781	157	10	abelian	abelian	ADJ
ejpam-3781	157	11	)	)	PUNCT
ejpam-3781	157	12	group	group	NOUN
ejpam-3781	157	13	.	.	PUNCT
ejpam-3781	158	1	let	let	VERB
ejpam-3781	158	2	n	n	NOUN
ejpam-3781	158	3	=	=	SYM
ejpam-3781	158	4	(	(	PUNCT
ejpam-3781	158	5	end(g),+	end(g),+	NOUN
ejpam-3781	158	6	,	,	PUNCT
ejpam-3781	158	7	◦	◦	NOUN
ejpam-3781	158	8	)	)	PUNCT
ejpam-3781	158	9	.	.	PUNCT
ejpam-3781	159	1	for	for	ADP
ejpam-3781	159	2	f	f	PROPN
ejpam-3781	159	3	in	in	ADP
ejpam-3781	159	4	n	n	PROPN
ejpam-3781	159	5	and	and	CCONJ
ejpam-3781	159	6	a	a	PRON
ejpam-3781	159	7	in	in	ADP
ejpam-3781	159	8	g	g	NOUN
ejpam-3781	159	9	,	,	PUNCT
ejpam-3781	159	10	define	define	VERB
ejpam-3781	159	11	fa	fa	NOUN
ejpam-3781	159	12	=	=	SYM
ejpam-3781	159	13	f(a	f(a	PROPN
ejpam-3781	159	14	)	)	PUNCT
ejpam-3781	159	15	.	.	PUNCT
ejpam-3781	160	1	then	then	ADV
ejpam-3781	160	2	g	g	PROPN
ejpam-3781	160	3	is	be	AUX
ejpam-3781	160	4	a	a	DET
ejpam-3781	160	5	modified	modify	VERB
ejpam-3781	160	6	near	near	ADP
ejpam-3781	160	7	module	module	NOUN
ejpam-3781	160	8	over	over	ADP
ejpam-3781	160	9	n	n	PROPN
ejpam-3781	160	10	.	.	PUNCT
ejpam-3781	161	1	define	define	VERB
ejpam-3781	161	2	α	α	NOUN
ejpam-3781	161	3	:	:	PUNCT
ejpam-3781	161	4	g→	g→	NOUN
ejpam-3781	161	5	n	n	X
ejpam-3781	161	6	by	by	ADP
ejpam-3781	161	7	α(a	α(a	NOUN
ejpam-3781	161	8	)	)	PUNCT
ejpam-3781	161	9	=	=	PUNCT
ejpam-3781	162	1	la	la	X
ejpam-3781	162	2	where	where	SCONJ
ejpam-3781	162	3	la	la	PROPN
ejpam-3781	162	4	is	be	AUX
ejpam-3781	162	5	the	the	DET
ejpam-3781	162	6	left	left	ADJ
ejpam-3781	162	7	addition	addition	NOUN
ejpam-3781	162	8	by	by	ADP
ejpam-3781	162	9	a	a	DET
ejpam-3781	162	10	:	:	PUNCT
ejpam-3781	162	11	la(x	la(x	NOUN
ejpam-3781	162	12	)	)	PUNCT
ejpam-3781	162	13	=	=	SYM
ejpam-3781	162	14	a+	a+	PUNCT
ejpam-3781	162	15	x	x	X
ejpam-3781	162	16	for	for	ADP
ejpam-3781	162	17	all	all	DET
ejpam-3781	162	18	x	x	SYM
ejpam-3781	162	19	∈	∈	PROPN
ejpam-3781	162	20	g.	g.	NOUN
ejpam-3781	162	21	then	then	ADV
ejpam-3781	162	22	α	α	INTJ
ejpam-3781	162	23	:	:	PUNCT
ejpam-3781	162	24	m	m	VERB
ejpam-3781	162	25	→	→	SYM
ejpam-3781	162	26	n	n	X
ejpam-3781	162	27	is	be	AUX
ejpam-3781	162	28	semilinear	semilinear	ADJ
ejpam-3781	162	29	.	.	PUNCT
ejpam-3781	162	30	example	example	NOUN
ejpam-3781	163	1	7	7	NUM
ejpam-3781	163	2	.	.	PUNCT
ejpam-3781	164	1	let	let	AUX
ejpam-3781	164	2	(	(	PUNCT
ejpam-3781	164	3	c,+	c,+	NOUN
ejpam-3781	164	4	)	)	PUNCT
ejpam-3781	164	5	be	be	VERB
ejpam-3781	164	6	the	the	DET
ejpam-3781	164	7	module	module	NOUN
ejpam-3781	164	8	of	of	ADP
ejpam-3781	164	9	complex	complex	ADJ
ejpam-3781	164	10	numbers	number	NOUN
ejpam-3781	164	11	over	over	ADP
ejpam-3781	164	12	the	the	DET
ejpam-3781	164	13	real	real	ADJ
ejpam-3781	164	14	field	field	NOUN
ejpam-3781	164	15	(	(	PUNCT
ejpam-3781	164	16	r,+	r,+	NUM
ejpam-3781	164	17	,	,	PUNCT
ejpam-3781	164	18	·	·	PUNCT
ejpam-3781	164	19	)	)	PUNCT
ejpam-3781	164	20	with	with	ADP
ejpam-3781	164	21	usual	usual	ADJ
ejpam-3781	164	22	product	product	NOUN
ejpam-3781	164	23	.	.	PUNCT
ejpam-3781	165	1	(	(	PUNCT
ejpam-3781	165	2	i	i	NOUN
ejpam-3781	165	3	)	)	PUNCT
ejpam-3781	165	4	define	define	VERB
ejpam-3781	165	5	f	f	NOUN
ejpam-3781	165	6	:	:	PUNCT
ejpam-3781	165	7	c→	c→	PUNCT
ejpam-3781	165	8	r	r	VERB
ejpam-3781	165	9	by	by	ADP
ejpam-3781	165	10	f(x+	f(x+	NOUN
ejpam-3781	165	11	iy	iy	INTJ
ejpam-3781	165	12	)	)	PUNCT
ejpam-3781	165	13	=	=	PRON
ejpam-3781	166	1	(	(	PUNCT
ejpam-3781	166	2	x2	x2	PROPN
ejpam-3781	166	3	+	+	CCONJ
ejpam-3781	166	4	y2	y2	NOUN
ejpam-3781	166	5	)	)	PUNCT
ejpam-3781	166	6	1	1	NUM
ejpam-3781	166	7	2	2	NUM
ejpam-3781	166	8	for	for	ADP
ejpam-3781	166	9	all	all	DET
ejpam-3781	166	10	x+	x+	PROPN
ejpam-3781	166	11	iy	iy	PROPN
ejpam-3781	166	12	∈	∈	PROPN
ejpam-3781	166	13	c.	c.	NOUN
ejpam-3781	166	14	for	for	ADP
ejpam-3781	166	15	any	any	DET
ejpam-3781	166	16	x1	x1	PROPN
ejpam-3781	166	17	+	+	CCONJ
ejpam-3781	166	18	iy1	iy1	NOUN
ejpam-3781	166	19	,	,	PUNCT
ejpam-3781	166	20	x2	x2	PROPN
ejpam-3781	166	21	+	+	PUNCT
ejpam-3781	166	22	iy2	iy2	PROPN
ejpam-3781	166	23	∈	∈	PROPN
ejpam-3781	166	24	c	c	NOUN
ejpam-3781	166	25	,	,	PUNCT
ejpam-3781	166	26	f(f(x1	f(f(x1	ADJ
ejpam-3781	166	27	+	+	NUM
ejpam-3781	166	28	iy1)(x2	iy1)(x2	NOUN
ejpam-3781	166	29	+	+	CCONJ
ejpam-3781	166	30	iy2	iy2	NOUN
ejpam-3781	166	31	)	)	PUNCT
ejpam-3781	166	32	)	)	PUNCT
ejpam-3781	167	1	=	=	PRON
ejpam-3781	167	2	f((x1	f((x1	NOUN
ejpam-3781	167	3	2	2	NUM
ejpam-3781	167	4	+	+	CCONJ
ejpam-3781	167	5	y1	y1	NOUN
ejpam-3781	167	6	2	2	NUM
ejpam-3781	167	7	)	)	PUNCT
ejpam-3781	167	8	1	1	NUM
ejpam-3781	167	9	2	2	NUM
ejpam-3781	167	10	(	(	PUNCT
ejpam-3781	167	11	x2	x2	NOUN
ejpam-3781	167	12	+	+	CCONJ
ejpam-3781	167	13	iy2	iy2	NOUN
ejpam-3781	167	14	)	)	PUNCT
ejpam-3781	167	15	)	)	PUNCT
ejpam-3781	168	1	=	=	PRON
ejpam-3781	168	2	f((x1	f((x1	NOUN
ejpam-3781	168	3	2	2	NUM
ejpam-3781	168	4	+	+	CCONJ
ejpam-3781	168	5	y1	y1	NOUN
ejpam-3781	168	6	2	2	NUM
ejpam-3781	168	7	)	)	PUNCT
ejpam-3781	168	8	1	1	NUM
ejpam-3781	168	9	2x2	2x2	NUM
ejpam-3781	168	10	+	+	CCONJ
ejpam-3781	168	11	i(x1	i(x1	ADJ
ejpam-3781	168	12	2	2	NUM
ejpam-3781	168	13	+	+	NUM
ejpam-3781	168	14	y1	y1	NOUN
ejpam-3781	168	15	2	2	NUM
ejpam-3781	168	16	)	)	PUNCT
ejpam-3781	168	17	1	1	NUM
ejpam-3781	168	18	2	2	NUM
ejpam-3781	168	19	y2	y2	NOUN
ejpam-3781	168	20	)	)	PUNCT
ejpam-3781	168	21	=	=	PUNCT
ejpam-3781	169	1	[	[	X
ejpam-3781	169	2	(	(	PUNCT
ejpam-3781	169	3	(	(	PUNCT
ejpam-3781	169	4	x1	x1	PROPN
ejpam-3781	169	5	2	2	NUM
ejpam-3781	169	6	+	+	NUM
ejpam-3781	169	7	y1	y1	NOUN
ejpam-3781	169	8	2	2	NUM
ejpam-3781	169	9	)	)	PUNCT
ejpam-3781	169	10	1	1	NUM
ejpam-3781	169	11	2x2	2x2	NUM
ejpam-3781	169	12	)	)	PUNCT
ejpam-3781	169	13	2	2	NUM
ejpam-3781	169	14	+	+	CCONJ
ejpam-3781	169	15	(	(	PUNCT
ejpam-3781	169	16	(	(	PUNCT
ejpam-3781	169	17	x1	x1	PROPN
ejpam-3781	169	18	2	2	NUM
ejpam-3781	169	19	+	+	NUM
ejpam-3781	169	20	y1	y1	NOUN
ejpam-3781	169	21	2	2	NUM
ejpam-3781	169	22	)	)	PUNCT
ejpam-3781	169	23	1	1	NUM
ejpam-3781	169	24	2	2	NUM
ejpam-3781	169	25	y2	y2	NOUN
ejpam-3781	169	26	)	)	PUNCT
ejpam-3781	169	27	2	2	NUM
ejpam-3781	169	28	]	]	SYM
ejpam-3781	169	29	1	1	NUM
ejpam-3781	169	30	2	2	NUM
ejpam-3781	169	31	=	=	SYM
ejpam-3781	169	32	(	(	PUNCT
ejpam-3781	169	33	x1	x1	NOUN
ejpam-3781	169	34	2	2	NUM
ejpam-3781	169	35	+	+	NUM
ejpam-3781	169	36	y1	y1	NOUN
ejpam-3781	169	37	2	2	NUM
ejpam-3781	169	38	)	)	PUNCT
ejpam-3781	169	39	1	1	NUM
ejpam-3781	169	40	2	2	NUM
ejpam-3781	169	41	(	(	PUNCT
ejpam-3781	169	42	x2	x2	NOUN
ejpam-3781	169	43	2	2	NUM
ejpam-3781	169	44	+	+	CCONJ
ejpam-3781	169	45	y2	y2	NOUN
ejpam-3781	169	46	2	2	NUM
ejpam-3781	169	47	)	)	PUNCT
ejpam-3781	169	48	1	1	NUM
ejpam-3781	169	49	2	2	NUM
ejpam-3781	169	50	=	=	NOUN
ejpam-3781	169	51	f(x1	f(x1	NOUN
ejpam-3781	169	52	+	+	NUM
ejpam-3781	169	53	iy1)f(x2	iy1)f(x2	NOUN
ejpam-3781	169	54	+	+	CCONJ
ejpam-3781	169	55	iy2	iy2	NOUN
ejpam-3781	169	56	)	)	PUNCT
ejpam-3781	169	57	.	.	PUNCT
ejpam-3781	170	1	hence	hence	ADV
ejpam-3781	170	2	f	f	PROPN
ejpam-3781	170	3	is	be	AUX
ejpam-3781	170	4	semilinear	semilinear	ADJ
ejpam-3781	170	5	;	;	PUNCT
ejpam-3781	170	6	therefore	therefore	ADV
ejpam-3781	170	7	(	(	PUNCT
ejpam-3781	170	8	c,+	c,+	NOUN
ejpam-3781	170	9	,	,	PUNCT
ejpam-3781	170	10	∗	∗	NOUN
ejpam-3781	170	11	)	)	PUNCT
ejpam-3781	170	12	is	be	AUX
ejpam-3781	170	13	a	a	DET
ejpam-3781	170	14	near	near	ADJ
ejpam-3781	170	15	ring	ring	NOUN
ejpam-3781	170	16	with	with	ADP
ejpam-3781	170	17	∗	∗	NOUN
ejpam-3781	170	18	defined	define	VERB
ejpam-3781	170	19	by	by	ADP
ejpam-3781	170	20	(	(	PUNCT
ejpam-3781	170	21	x1	x1	PROPN
ejpam-3781	170	22	+	+	NUM
ejpam-3781	170	23	iy1	iy1	NOUN
ejpam-3781	170	24	)	)	PUNCT
ejpam-3781	170	25	∗	∗	NOUN
ejpam-3781	170	26	(	(	PUNCT
ejpam-3781	170	27	x2	x2	NOUN
ejpam-3781	170	28	+	+	NUM
ejpam-3781	170	29	iy2	iy2	NOUN
ejpam-3781	170	30	)	)	PUNCT
ejpam-3781	170	31	=	=	SYM
ejpam-3781	170	32	f(x2	f(x2	NOUN
ejpam-3781	170	33	+	+	X
ejpam-3781	170	34	iy2)(x1	iy2)(x1	ADJ
ejpam-3781	170	35	+	+	CCONJ
ejpam-3781	170	36	iy1	iy1	NOUN
ejpam-3781	170	37	)	)	PUNCT
ejpam-3781	170	38	=	=	PUNCT
ejpam-3781	171	1	(	(	PUNCT
ejpam-3781	171	2	x2	x2	NOUN
ejpam-3781	171	3	2	2	NUM
ejpam-3781	171	4	+	+	CCONJ
ejpam-3781	171	5	y2	y2	NOUN
ejpam-3781	171	6	2	2	NUM
ejpam-3781	171	7	)	)	PUNCT
ejpam-3781	171	8	1	1	NUM
ejpam-3781	171	9	2	2	NUM
ejpam-3781	171	10	(	(	PUNCT
ejpam-3781	171	11	x1	x1	PROPN
ejpam-3781	171	12	+	+	NUM
ejpam-3781	171	13	iy1	iy1	NOUN
ejpam-3781	171	14	)	)	PUNCT
ejpam-3781	171	15	.	.	PUNCT
ejpam-3781	172	1	a.v	a.v	PROPN
ejpam-3781	172	2	.	.	PROPN
ejpam-3781	172	3	ramakrishna	ramakrishna	PROPN
ejpam-3781	172	4	,	,	PUNCT
ejpam-3781	172	5	t.v.n	t.v.n	PROPN
ejpam-3781	172	6	.	.	PUNCT
ejpam-3781	173	1	prasanna	prasanna	PROPN
ejpam-3781	173	2	,	,	PUNCT
ejpam-3781	173	3	d.v	d.v	PROPN
ejpam-3781	173	4	.	.	PROPN
ejpam-3781	173	5	lakshmi	lakshmi	PROPN
ejpam-3781	173	6	/	/	SYM
ejpam-3781	173	7	eur	eur	PROPN
ejpam-3781	173	8	.	.	PUNCT
ejpam-3781	174	1	j.	j.	PROPN
ejpam-3781	174	2	pure	pure	PROPN
ejpam-3781	174	3	appl	appl	PROPN
ejpam-3781	174	4	.	.	PROPN
ejpam-3781	174	5	math	math	PROPN
ejpam-3781	174	6	,	,	PUNCT
ejpam-3781	174	7	14	14	NUM
ejpam-3781	174	8	(	(	PUNCT
ejpam-3781	174	9	1	1	NUM
ejpam-3781	174	10	)	)	PUNCT
ejpam-3781	174	11	(	(	PUNCT
ejpam-3781	174	12	2021	2021	NUM
ejpam-3781	174	13	)	)	PUNCT
ejpam-3781	174	14	,	,	PUNCT
ejpam-3781	174	15	126	126	NUM
ejpam-3781	174	16	-	-	SYM
ejpam-3781	174	17	134	134	NUM
ejpam-3781	174	18	131	131	NUM
ejpam-3781	174	19	(	(	PUNCT
ejpam-3781	174	20	ii	ii	NOUN
ejpam-3781	174	21	)	)	PUNCT
ejpam-3781	174	22	define	define	VERB
ejpam-3781	174	23	f	f	PROPN
ejpam-3781	174	24	:	:	PUNCT
ejpam-3781	174	25	c→	c→	PUNCT
ejpam-3781	174	26	r	r	VERB
ejpam-3781	174	27	by	by	ADP
ejpam-3781	174	28	f(x+	f(x+	NOUN
ejpam-3781	174	29	iy	iy	PRON
ejpam-3781	174	30	)	)	PUNCT
ejpam-3781	175	1	=	=	SYM
ejpam-3781	175	2	|x|	|x|	PROPN
ejpam-3781	175	3	for	for	ADP
ejpam-3781	175	4	all	all	DET
ejpam-3781	175	5	x+	x+	PROPN
ejpam-3781	175	6	iy	iy	PROPN
ejpam-3781	175	7	∈	∈	PROPN
ejpam-3781	175	8	c.	c.	NOUN
ejpam-3781	175	9	for	for	ADP
ejpam-3781	175	10	any	any	DET
ejpam-3781	175	11	x1	x1	PROPN
ejpam-3781	175	12	+	+	CCONJ
ejpam-3781	175	13	iy1	iy1	NOUN
ejpam-3781	175	14	,	,	PUNCT
ejpam-3781	175	15	x2	x2	PROPN
ejpam-3781	176	1	+	+	PUNCT
ejpam-3781	176	2	iy2	iy2	PROPN
ejpam-3781	176	3	∈	∈	PROPN
ejpam-3781	176	4	c	c	NOUN
ejpam-3781	176	5	,	,	PUNCT
ejpam-3781	176	6	f(f(x1	f(f(x1	ADJ
ejpam-3781	176	7	+	+	NUM
ejpam-3781	176	8	iy1)(x2	iy1)(x2	NOUN
ejpam-3781	176	9	+	+	CCONJ
ejpam-3781	176	10	iy2	iy2	NOUN
ejpam-3781	176	11	)	)	PUNCT
ejpam-3781	176	12	)	)	PUNCT
ejpam-3781	177	1	=	=	SYM
ejpam-3781	177	2	f(|x1|(x2	f(|x1|(x2	X
ejpam-3781	178	1	+	+	CCONJ
ejpam-3781	178	2	iy2	iy2	NOUN
ejpam-3781	178	3	)	)	PUNCT
ejpam-3781	178	4	)	)	PUNCT
ejpam-3781	179	1	=	=	SYM
ejpam-3781	179	2	f(|x1|x2	f(|x1|x2	NOUN
ejpam-3781	179	3	+	+	CCONJ
ejpam-3781	179	4	i|x1|y2	i|x1|y2	NOUN
ejpam-3781	179	5	)	)	PUNCT
ejpam-3781	179	6	=	=	SYM
ejpam-3781	179	7	||x1|x2|	||x1|x2|	PROPN
ejpam-3781	179	8	=	=	SYM
ejpam-3781	179	9	|x1||x2|	|x1||x2|	PROPN
ejpam-3781	179	10	=	=	PUNCT
ejpam-3781	180	1	f(x1	f(x1	PROPN
ejpam-3781	180	2	+	+	NUM
ejpam-3781	180	3	iy1)f(x2	iy1)f(x2	NOUN
ejpam-3781	180	4	+	+	CCONJ
ejpam-3781	180	5	iy2	iy2	NOUN
ejpam-3781	180	6	)	)	PUNCT
ejpam-3781	180	7	.	.	PUNCT
ejpam-3781	181	1	hence	hence	ADV
ejpam-3781	181	2	f	f	PROPN
ejpam-3781	181	3	is	be	AUX
ejpam-3781	181	4	semilinear	semilinear	ADJ
ejpam-3781	181	5	;	;	PUNCT
ejpam-3781	181	6	therefore	therefore	ADV
ejpam-3781	181	7	(	(	PUNCT
ejpam-3781	181	8	c,+	c,+	NOUN
ejpam-3781	181	9	,	,	PUNCT
ejpam-3781	181	10	∗	∗	NOUN
ejpam-3781	181	11	)	)	PUNCT
ejpam-3781	181	12	is	be	AUX
ejpam-3781	181	13	a	a	DET
ejpam-3781	181	14	near	near	ADJ
ejpam-3781	181	15	ring	ring	NOUN
ejpam-3781	181	16	with	with	ADP
ejpam-3781	181	17	∗	∗	NOUN
ejpam-3781	181	18	defined	define	VERB
ejpam-3781	181	19	by	by	ADP
ejpam-3781	181	20	(	(	PUNCT
ejpam-3781	181	21	x1	x1	PROPN
ejpam-3781	181	22	+	+	NUM
ejpam-3781	181	23	iy1	iy1	NOUN
ejpam-3781	181	24	)	)	PUNCT
ejpam-3781	181	25	∗	∗	NOUN
ejpam-3781	181	26	(	(	PUNCT
ejpam-3781	181	27	x2	x2	NOUN
ejpam-3781	181	28	+	+	NUM
ejpam-3781	181	29	iy2	iy2	NOUN
ejpam-3781	181	30	)	)	PUNCT
ejpam-3781	181	31	=	=	SYM
ejpam-3781	181	32	f(x2	f(x2	NOUN
ejpam-3781	181	33	+	+	X
ejpam-3781	181	34	iy2)(x1	iy2)(x1	ADJ
ejpam-3781	181	35	+	+	CCONJ
ejpam-3781	181	36	iy1	iy1	NOUN
ejpam-3781	181	37	)	)	PUNCT
ejpam-3781	181	38	=	=	SYM
ejpam-3781	182	1	|x2|(x1	|x2|(x1	PROPN
ejpam-3781	182	2	+	+	NUM
ejpam-3781	182	3	iy1	iy1	NOUN
ejpam-3781	182	4	)	)	PUNCT
ejpam-3781	182	5	.	.	PUNCT
ejpam-3781	183	1	example	example	NOUN
ejpam-3781	184	1	8	8	NUM
ejpam-3781	184	2	.	.	PUNCT
ejpam-3781	185	1	let	let	VERB
ejpam-3781	185	2	m	m	VERB
ejpam-3781	185	3	=	=	VERB
ejpam-3781	185	4	{	{	PUNCT
ejpam-3781	185	5	a	a	DET
ejpam-3781	185	6	+	+	NOUN
ejpam-3781	185	7	bi	bi	NOUN
ejpam-3781	185	8	+	+	CCONJ
ejpam-3781	185	9	cj	cj	PROPN
ejpam-3781	185	10	+	+	X
ejpam-3781	185	11	dk|a	dk|a	PROPN
ejpam-3781	185	12	,	,	PUNCT
ejpam-3781	185	13	b	b	PROPN
ejpam-3781	185	14	,	,	PUNCT
ejpam-3781	185	15	c	c	NOUN
ejpam-3781	185	16	,	,	PUNCT
ejpam-3781	185	17	d	d	PROPN
ejpam-3781	185	18	∈	∈	PROPN
ejpam-3781	185	19	r	r	AUX
ejpam-3781	185	20	}	}	PUNCT
ejpam-3781	185	21	be	be	AUX
ejpam-3781	185	22	the	the	DET
ejpam-3781	185	23	ring	ring	NOUN
ejpam-3781	185	24	of	of	ADP
ejpam-3781	185	25	real	real	ADJ
ejpam-3781	185	26	quaternions	quaternion	NOUN
ejpam-3781	185	27	.	.	PUNCT
ejpam-3781	186	1	then	then	ADV
ejpam-3781	186	2	m	m	PROPN
ejpam-3781	186	3	is	be	AUX
ejpam-3781	186	4	a	a	DET
ejpam-3781	186	5	modified	modify	VERB
ejpam-3781	186	6	near	near	ADP
ejpam-3781	186	7	module	module	NOUN
ejpam-3781	186	8	over	over	ADP
ejpam-3781	186	9	the	the	DET
ejpam-3781	186	10	real	real	ADJ
ejpam-3781	186	11	number	number	NOUN
ejpam-3781	186	12	field	field	NOUN
ejpam-3781	186	13	(	(	PUNCT
ejpam-3781	186	14	r,+	r,+	NUM
ejpam-3781	186	15	,	,	PUNCT
ejpam-3781	186	16	·	·	PUNCT
ejpam-3781	186	17	)	)	PUNCT
ejpam-3781	186	18	.	.	PUNCT
ejpam-3781	187	1	define	define	VERB
ejpam-3781	187	2	f	f	PROPN
ejpam-3781	187	3	:	:	PUNCT
ejpam-3781	187	4	m	m	VERB
ejpam-3781	187	5	→	→	SYM
ejpam-3781	187	6	r	r	AUX
ejpam-3781	187	7	by	by	ADP
ejpam-3781	187	8	f(a+	f(a+	PROPN
ejpam-3781	187	9	bi+	bi+	PROPN
ejpam-3781	187	10	cj	cj	PROPN
ejpam-3781	188	1	+	+	CCONJ
ejpam-3781	188	2	dk	dk	PROPN
ejpam-3781	188	3	)	)	PUNCT
ejpam-3781	188	4	=	=	SYM
ejpam-3781	188	5	a2	a2	PROPN
ejpam-3781	188	6	+	+	CCONJ
ejpam-3781	188	7	b2	b2	NOUN
ejpam-3781	188	8	+	+	CCONJ
ejpam-3781	188	9	c2	c2	PROPN
ejpam-3781	188	10	+	+	CCONJ
ejpam-3781	188	11	d2	d2	PROPN
ejpam-3781	188	12	for	for	ADP
ejpam-3781	188	13	all	all	PRON
ejpam-3781	188	14	a+	a+	PRON
ejpam-3781	188	15	bi+	bi+	PROPN
ejpam-3781	188	16	cj	cj	PROPN
ejpam-3781	189	1	+	+	CCONJ
ejpam-3781	189	2	dk	dk	PROPN
ejpam-3781	189	3	∈m	∈m	NOUN
ejpam-3781	189	4	.	.	PUNCT
ejpam-3781	190	1	for	for	ADP
ejpam-3781	190	2	any	any	DET
ejpam-3781	190	3	a1	a1	NOUN
ejpam-3781	190	4	+	+	CCONJ
ejpam-3781	190	5	b1i+	b1i+	NOUN
ejpam-3781	190	6	c1j	c1j	NOUN
ejpam-3781	190	7	+	+	CCONJ
ejpam-3781	190	8	d1k	d1k	PROPN
ejpam-3781	190	9	,	,	PUNCT
ejpam-3781	190	10	a2	a2	PROPN
ejpam-3781	190	11	+	+	CCONJ
ejpam-3781	190	12	b2i+	b2i+	VERB
ejpam-3781	190	13	c2j	c2j	PROPN
ejpam-3781	190	14	+	+	CCONJ
ejpam-3781	190	15	d2k	d2k	PROPN
ejpam-3781	190	16	∈m	∈m	NOUN
ejpam-3781	190	17	,	,	PUNCT
ejpam-3781	190	18	f(f(a1	f(f(a1	NOUN
ejpam-3781	190	19	+	+	CCONJ
ejpam-3781	190	20	b1i+	b1i+	NOUN
ejpam-3781	190	21	c1j	c1j	NOUN
ejpam-3781	190	22	+	+	NOUN
ejpam-3781	190	23	d1k)(a2	d1k)(a2	NOUN
ejpam-3781	190	24	+	+	CCONJ
ejpam-3781	190	25	b2i+	b2i+	VERB
ejpam-3781	190	26	c2j	c2j	PROPN
ejpam-3781	190	27	+	+	CCONJ
ejpam-3781	190	28	d2k	d2k	NOUN
ejpam-3781	190	29	)	)	PUNCT
ejpam-3781	190	30	)	)	PUNCT
ejpam-3781	191	1	=	=	PRON
ejpam-3781	191	2	f((a1	f((a1	VERB
ejpam-3781	191	3	2	2	NUM
ejpam-3781	191	4	+	+	CCONJ
ejpam-3781	191	5	b1	b1	NOUN
ejpam-3781	191	6	2	2	NUM
ejpam-3781	191	7	+	+	CCONJ
ejpam-3781	191	8	c1	c1	NOUN
ejpam-3781	191	9	2	2	NUM
ejpam-3781	191	10	+	+	CCONJ
ejpam-3781	191	11	d1	d1	PROPN
ejpam-3781	191	12	2)(a2	2)(a2	NOUN
ejpam-3781	192	1	+	+	CCONJ
ejpam-3781	192	2	b2i+	b2i+	NOUN
ejpam-3781	192	3	c2j	c2j	PROPN
ejpam-3781	192	4	+	+	CCONJ
ejpam-3781	192	5	d2k	d2k	NOUN
ejpam-3781	192	6	)	)	PUNCT
ejpam-3781	192	7	)	)	PUNCT
ejpam-3781	193	1	=	=	PRON
ejpam-3781	193	2	f(a1	f(a1	NOUN
ejpam-3781	193	3	+	+	CCONJ
ejpam-3781	193	4	b1i+	b1i+	NOUN
ejpam-3781	193	5	c1j	c1j	VERB
ejpam-3781	193	6	+	+	CCONJ
ejpam-3781	193	7	d1k)f(a2	d1k)f(a2	ADJ
ejpam-3781	193	8	+	+	CCONJ
ejpam-3781	193	9	b2i+	b2i+	NOUN
ejpam-3781	193	10	c2j	c2j	PROPN
ejpam-3781	193	11	+	+	CCONJ
ejpam-3781	193	12	d2k	d2k	NOUN
ejpam-3781	193	13	)	)	PUNCT
ejpam-3781	193	14	.	.	PUNCT
ejpam-3781	194	1	hence	hence	ADV
ejpam-3781	194	2	f	f	PROPN
ejpam-3781	194	3	is	be	AUX
ejpam-3781	194	4	semilinear	semilinear	ADJ
ejpam-3781	194	5	and	and	CCONJ
ejpam-3781	194	6	therefore	therefore	ADV
ejpam-3781	194	7	(	(	PUNCT
ejpam-3781	194	8	m,+	m,+	INTJ
ejpam-3781	194	9	,	,	PUNCT
ejpam-3781	194	10	∗	∗	NOUN
ejpam-3781	194	11	)	)	PUNCT
ejpam-3781	194	12	is	be	AUX
ejpam-3781	194	13	a	a	DET
ejpam-3781	194	14	near	near	ADJ
ejpam-3781	194	15	ring	ring	NOUN
ejpam-3781	194	16	with	with	ADP
ejpam-3781	194	17	(	(	PUNCT
ejpam-3781	194	18	a1	a1	NOUN
ejpam-3781	194	19	+	+	CCONJ
ejpam-3781	194	20	b1i+	b1i+	NOUN
ejpam-3781	194	21	c1j	c1j	NOUN
ejpam-3781	194	22	+	+	CCONJ
ejpam-3781	194	23	d1k	d1k	NOUN
ejpam-3781	194	24	)	)	PUNCT
ejpam-3781	194	25	∗	∗	NOUN
ejpam-3781	194	26	(	(	PUNCT
ejpam-3781	194	27	a2	a2	NOUN
ejpam-3781	194	28	+	+	CCONJ
ejpam-3781	194	29	b2i+	b2i+	VERB
ejpam-3781	194	30	c2j	c2j	PROPN
ejpam-3781	194	31	+	+	CCONJ
ejpam-3781	194	32	d2k	d2k	NOUN
ejpam-3781	194	33	)	)	PUNCT
ejpam-3781	194	34	=	=	PUNCT
ejpam-3781	194	35	f(a2	f(a2	VERB
ejpam-3781	195	1	+	+	CCONJ
ejpam-3781	195	2	b2i+	b2i+	NOUN
ejpam-3781	195	3	c2j	c2j	NOUN
ejpam-3781	195	4	+	+	CCONJ
ejpam-3781	195	5	d2k)(a1	d2k)(a1	X
ejpam-3781	195	6	+	+	PUNCT
ejpam-3781	195	7	b1i+	b1i+	NOUN
ejpam-3781	195	8	c1j	c1j	NOUN
ejpam-3781	195	9	+	+	CCONJ
ejpam-3781	195	10	d1k	d1k	NOUN
ejpam-3781	195	11	)	)	PUNCT
ejpam-3781	195	12	=	=	SYM
ejpam-3781	195	13	(	(	PUNCT
ejpam-3781	195	14	a2	a2	PROPN
ejpam-3781	195	15	2	2	NUM
ejpam-3781	195	16	+	+	NOUN
ejpam-3781	195	17	b2	b2	NOUN
ejpam-3781	195	18	2	2	NUM
ejpam-3781	195	19	+	+	CCONJ
ejpam-3781	195	20	c2	c2	PROPN
ejpam-3781	195	21	2	2	NUM
ejpam-3781	195	22	+	+	CCONJ
ejpam-3781	195	23	d2	d2	PROPN
ejpam-3781	195	24	2)(a1	2)(a1	NUM
ejpam-3781	195	25	+	+	NUM
ejpam-3781	195	26	b1i+	b1i+	NOUN
ejpam-3781	195	27	c1j	c1j	NOUN
ejpam-3781	195	28	+	+	NUM
ejpam-3781	195	29	d1k	d1k	NOUN
ejpam-3781	195	30	)	)	PUNCT
ejpam-3781	195	31	.	.	PUNCT
ejpam-3781	196	1	example	example	NOUN
ejpam-3781	197	1	9	9	NUM
ejpam-3781	197	2	.	.	PUNCT
ejpam-3781	198	1	let	let	VERB
ejpam-3781	198	2	m	m	PRON
ejpam-3781	198	3	be	be	AUX
ejpam-3781	198	4	the	the	DET
ejpam-3781	198	5	set	set	NOUN
ejpam-3781	198	6	of	of	ADP
ejpam-3781	198	7	all	all	DET
ejpam-3781	198	8	n×	n×	PROPN
ejpam-3781	198	9	n	n	PRON
ejpam-3781	198	10	real	real	ADJ
ejpam-3781	198	11	matrices	matrix	NOUN
ejpam-3781	198	12	.	.	PUNCT
ejpam-3781	199	1	then	then	ADV
ejpam-3781	199	2	(	(	PUNCT
ejpam-3781	199	3	m,+	m,+	INTJ
ejpam-3781	199	4	,	,	PUNCT
ejpam-3781	199	5	·	·	PUNCT
ejpam-3781	199	6	)	)	PUNCT
ejpam-3781	199	7	is	be	AUX
ejpam-3781	199	8	a	a	DET
ejpam-3781	199	9	strong	strong	ADJ
ejpam-3781	199	10	near	near	ADJ
ejpam-3781	199	11	module	module	NOUN
ejpam-3781	199	12	over	over	ADP
ejpam-3781	199	13	the	the	DET
ejpam-3781	199	14	real	real	ADJ
ejpam-3781	199	15	number	number	NOUN
ejpam-3781	199	16	field	field	NOUN
ejpam-3781	199	17	(	(	PUNCT
ejpam-3781	199	18	r,+	r,+	NUM
ejpam-3781	199	19	,	,	PUNCT
ejpam-3781	199	20	·	·	PUNCT
ejpam-3781	199	21	)	)	PUNCT
ejpam-3781	199	22	.	.	PUNCT
ejpam-3781	200	1	define	define	VERB
ejpam-3781	200	2	f	f	PROPN
ejpam-3781	200	3	:	:	PUNCT
ejpam-3781	200	4	m	m	VERB
ejpam-3781	200	5	→	→	SYM
ejpam-3781	200	6	r	r	NOUN
ejpam-3781	200	7	by	by	ADP
ejpam-3781	200	8	f(a	f(a	NOUN
ejpam-3781	200	9	)	)	PUNCT
ejpam-3781	201	1	=	=	PUNCT
ejpam-3781	201	2	∑	∑	PUNCT
ejpam-3781	201	3	1≤i	1≤i	INTJ
ejpam-3781	201	4	,	,	PUNCT
ejpam-3781	201	5	j≤n	j≤n	PROPN
ejpam-3781	201	6	(	(	PUNCT
ejpam-3781	201	7	aij	aij	PROPN
ejpam-3781	201	8	)	)	PUNCT
ejpam-3781	201	9	2	2	NUM
ejpam-3781	201	10	.	.	PUNCT
ejpam-3781	202	1	then	then	ADV
ejpam-3781	202	2	f	f	PROPN
ejpam-3781	202	3	is	be	AUX
ejpam-3781	202	4	a	a	DET
ejpam-3781	202	5	semilinear	semilinear	ADJ
ejpam-3781	202	6	map	map	NOUN
ejpam-3781	202	7	and	and	CCONJ
ejpam-3781	202	8	hence	hence	ADV
ejpam-3781	202	9	(	(	PUNCT
ejpam-3781	202	10	m,+	m,+	INTJ
ejpam-3781	202	11	,	,	PUNCT
ejpam-3781	202	12	∗	∗	NOUN
ejpam-3781	202	13	)	)	PUNCT
ejpam-3781	202	14	is	be	AUX
ejpam-3781	202	15	a	a	DET
ejpam-3781	202	16	near	near	ADJ
ejpam-3781	202	17	ring	ring	NOUN
ejpam-3781	202	18	with	with	ADP
ejpam-3781	202	19	a	a	DET
ejpam-3781	202	20	∗b	∗b	PROPN
ejpam-3781	202	21	=	=	SYM
ejpam-3781	202	22	f(b)a	f(b)a	PROPN
ejpam-3781	202	23	.	.	PUNCT
ejpam-3781	202	24	theorem	theorem	PROPN
ejpam-3781	202	25	5	5	NUM
ejpam-3781	202	26	.	.	PUNCT
ejpam-3781	203	1	let	let	AUX
ejpam-3781	203	2	(	(	PUNCT
ejpam-3781	203	3	m,+	m,+	INTJ
ejpam-3781	203	4	,	,	PUNCT
ejpam-3781	203	5	·	·	PUNCT
ejpam-3781	203	6	)	)	PUNCT
ejpam-3781	203	7	be	be	AUX
ejpam-3781	203	8	a	a	DET
ejpam-3781	203	9	modified	modify	VERB
ejpam-3781	203	10	near	near	ADP
ejpam-3781	203	11	module	module	NOUN
ejpam-3781	203	12	over	over	ADP
ejpam-3781	203	13	n	n	PROPN
ejpam-3781	203	14	.	.	PUNCT
ejpam-3781	204	1	define	define	VERB
ejpam-3781	204	2	the	the	DET
ejpam-3781	204	3	function	function	NOUN
ejpam-3781	204	4	�	�	PROPN
ejpam-3781	204	5	from	from	ADP
ejpam-3781	204	6	n	n	PROPN
ejpam-3781	204	7	×m	×m	NOUN
ejpam-3781	204	8	into	into	ADP
ejpam-3781	204	9	m	m	PROPN
ejpam-3781	204	10	as	as	ADP
ejpam-3781	204	11	�	�	PROPN
ejpam-3781	204	12	(	(	PUNCT
ejpam-3781	204	13	n	n	CCONJ
ejpam-3781	204	14	,	,	PUNCT
ejpam-3781	204	15	m	m	NOUN
ejpam-3781	204	16	)	)	PUNCT
ejpam-3781	204	17	=	=	SYM
ejpam-3781	204	18	n	n	CCONJ
ejpam-3781	204	19	�	�	PROPN
ejpam-3781	204	20	m	m	NOUN
ejpam-3781	204	21	=	=	NOUN
ejpam-3781	204	22	f(n)m	f(n)m	PROPN
ejpam-3781	204	23	for	for	ADP
ejpam-3781	204	24	all	all	DET
ejpam-3781	204	25	m	m	NOUN
ejpam-3781	204	26	∈m	∈m	NOUN
ejpam-3781	204	27	and	and	CCONJ
ejpam-3781	204	28	n	n	PRON
ejpam-3781	204	29	∈	∈	PROPN
ejpam-3781	204	30	n	n	NOUN
ejpam-3781	204	31	.	.	PUNCT
ejpam-3781	205	1	then	then	ADV
ejpam-3781	205	2	(	(	PUNCT
ejpam-3781	205	3	m,+	m,+	PROPN
ejpam-3781	205	4	,	,	PUNCT
ejpam-3781	205	5	�	�	PROPN
ejpam-3781	205	6	)	)	PUNCT
ejpam-3781	205	7	is	be	AUX
ejpam-3781	205	8	a	a	DET
ejpam-3781	205	9	modified	modify	VERB
ejpam-3781	205	10	near	near	ADP
ejpam-3781	205	11	module	module	NOUN
ejpam-3781	205	12	over	over	ADP
ejpam-3781	205	13	nf	nf	NOUN
ejpam-3781	205	14	,	,	PUNCT
ejpam-3781	205	15	where	where	SCONJ
ejpam-3781	205	16	nf	nf	PRON
ejpam-3781	205	17	is	be	AUX
ejpam-3781	205	18	a	a	DET
ejpam-3781	205	19	near	near	ADJ
ejpam-3781	205	20	ring	ring	NOUN
ejpam-3781	205	21	induced	induce	VERB
ejpam-3781	205	22	by	by	ADP
ejpam-3781	205	23	the	the	DET
ejpam-3781	205	24	semilinear	semilinear	PROPN
ejpam-3781	205	25	map	map	NOUN
ejpam-3781	206	1	f	f	PROPN
ejpam-3781	206	2	.	.	PUNCT
ejpam-3781	207	1	proof	proof	NOUN
ejpam-3781	207	2	.	.	PUNCT
ejpam-3781	208	1	for	for	ADP
ejpam-3781	208	2	any	any	DET
ejpam-3781	208	3	n	n	PRON
ejpam-3781	208	4	∈	∈	PROPN
ejpam-3781	208	5	n	n	NOUN
ejpam-3781	208	6	and	and	CCONJ
ejpam-3781	208	7	m1,m2	m1,m2	PROPN
ejpam-3781	208	8	∈m	∈m	NOUN
ejpam-3781	208	9	,	,	PUNCT
ejpam-3781	208	10	n	n	CCONJ
ejpam-3781	208	11	�	�	PROPN
ejpam-3781	208	12	(	(	PUNCT
ejpam-3781	208	13	m1	m1	PROPN
ejpam-3781	208	14	+	+	NOUN
ejpam-3781	208	15	m2	m2	PROPN
ejpam-3781	208	16	)	)	PUNCT
ejpam-3781	208	17	=	=	PUNCT
ejpam-3781	208	18	f(n)(m1	f(n)(m1	NOUN
ejpam-3781	208	19	+	+	NOUN
ejpam-3781	208	20	m2	m2	PROPN
ejpam-3781	208	21	)	)	PUNCT
ejpam-3781	209	1	=	=	SYM
ejpam-3781	209	2	f(n)m1	f(n)m1	NOUN
ejpam-3781	209	3	+	+	CCONJ
ejpam-3781	209	4	f(n)m2	f(n)m2	NOUN
ejpam-3781	209	5	=	=	SYM
ejpam-3781	209	6	n	n	CCONJ
ejpam-3781	209	7	�	�	PROPN
ejpam-3781	209	8	m1	m1	PROPN
ejpam-3781	209	9	+	+	CCONJ
ejpam-3781	209	10	n	n	CCONJ
ejpam-3781	209	11	�	�	PROPN
ejpam-3781	209	12	m2	m2	PROPN
ejpam-3781	209	13	.	.	PROPN
ejpam-3781	210	1	for	for	ADP
ejpam-3781	210	2	any	any	DET
ejpam-3781	210	3	n1	n1	NOUN
ejpam-3781	210	4	,	,	PUNCT
ejpam-3781	210	5	n2	n2	NOUN
ejpam-3781	210	6	∈	∈	PROPN
ejpam-3781	210	7	n	n	PRON
ejpam-3781	210	8	and	and	CCONJ
ejpam-3781	210	9	m	m	NOUN
ejpam-3781	210	10	∈m	∈m	NOUN
ejpam-3781	210	11	,	,	PUNCT
ejpam-3781	210	12	(	(	PUNCT
ejpam-3781	210	13	n1	n1	NOUN
ejpam-3781	210	14	∗	∗	NOUN
ejpam-3781	210	15	n2)	n2)	NUM
ejpam-3781	210	16	�	�	PROPN
ejpam-3781	210	17	m	m	NOUN
ejpam-3781	210	18	=	=	NOUN
ejpam-3781	210	19	f(n1	f(n1	NOUN
ejpam-3781	210	20	∗	∗	NOUN
ejpam-3781	210	21	n2)m	n2)m	NOUN
ejpam-3781	210	22	=	=	SYM
ejpam-3781	210	23	f(n1f(n2))m	f(n1f(n2))m	ADJ
ejpam-3781	210	24	=	=	PUNCT
ejpam-3781	211	1	[	[	X
ejpam-3781	211	2	f(n1)f(n2)]m	f(n1)f(n2)]m	PUNCT
ejpam-3781	211	3	and	and	CCONJ
ejpam-3781	211	4	n1	n1	PROPN
ejpam-3781	211	5	�	�	PROPN
ejpam-3781	211	6	(	(	PUNCT
ejpam-3781	211	7	n2	n2	PROPN
ejpam-3781	211	8	�	�	PROPN
ejpam-3781	211	9	m	m	NOUN
ejpam-3781	211	10	)	)	PUNCT
ejpam-3781	211	11	=	=	SYM
ejpam-3781	211	12	f(n1)(n2	f(n1)(n2	PROPN
ejpam-3781	211	13	�	�	PROPN
ejpam-3781	211	14	m	m	NOUN
ejpam-3781	211	15	)	)	PUNCT
ejpam-3781	211	16	=	=	SYM
ejpam-3781	211	17	f(n1)[f(n2)m	f(n1)[f(n2)m	NOUN
ejpam-3781	211	18	]	]	X
ejpam-3781	211	19	=	=	X
ejpam-3781	212	1	[	[	X
ejpam-3781	212	2	f(n1)f(n2)]m	f(n1)f(n2)]m	X
ejpam-3781	212	3	.	.	PUNCT
ejpam-3781	212	4	therefore	therefore	ADV
ejpam-3781	212	5	(	(	PUNCT
ejpam-3781	212	6	m,+	m,+	PROPN
ejpam-3781	212	7	,	,	PUNCT
ejpam-3781	212	8	�	�	PROPN
ejpam-3781	212	9	)	)	PUNCT
ejpam-3781	212	10	is	be	AUX
ejpam-3781	212	11	a	a	DET
ejpam-3781	212	12	modified	modify	VERB
ejpam-3781	212	13	near	near	ADP
ejpam-3781	212	14	module	module	NOUN
ejpam-3781	212	15	over	over	ADP
ejpam-3781	212	16	nf	nf	NOUN
ejpam-3781	212	17	.	.	PUNCT
ejpam-3781	213	1	theorem	theorem	ADJ
ejpam-3781	213	2	6	6	NUM
ejpam-3781	213	3	.	.	PUNCT
ejpam-3781	214	1	let	let	VERB
ejpam-3781	214	2	m1,m2	m1,m2	PROPN
ejpam-3781	214	3	be	be	AUX
ejpam-3781	214	4	modified	modify	VERB
ejpam-3781	214	5	near	near	ADP
ejpam-3781	214	6	modules	module	NOUN
ejpam-3781	214	7	over	over	ADP
ejpam-3781	214	8	n	n	PROPN
ejpam-3781	214	9	and	and	CCONJ
ejpam-3781	214	10	f	f	PROPN
ejpam-3781	214	11	:	:	PUNCT
ejpam-3781	214	12	m1	m1	PROPN
ejpam-3781	214	13	→	→	SYM
ejpam-3781	214	14	n	n	CCONJ
ejpam-3781	214	15	be	be	AUX
ejpam-3781	214	16	a	a	DET
ejpam-3781	214	17	semilinear	semilinear	NOUN
ejpam-3781	214	18	map	map	NOUN
ejpam-3781	214	19	and	and	CCONJ
ejpam-3781	214	20	φ	φ	NOUN
ejpam-3781	214	21	:	:	PUNCT
ejpam-3781	215	1	m2	m2	PROPN
ejpam-3781	215	2	→m1	→m1	X
ejpam-3781	215	3	be	be	AUX
ejpam-3781	215	4	a	a	DET
ejpam-3781	215	5	near	near	ADJ
ejpam-3781	215	6	module	module	NOUN
ejpam-3781	215	7	homomorphism	homomorphism	NOUN
ejpam-3781	215	8	.	.	PUNCT
ejpam-3781	216	1	then	then	ADV
ejpam-3781	216	2	foφ	foφ	PROPN
ejpam-3781	216	3	is	be	AUX
ejpam-3781	216	4	a	a	DET
ejpam-3781	216	5	semilinear	semilinear	NOUN
ejpam-3781	216	6	map	map	NOUN
ejpam-3781	216	7	.	.	PUNCT
ejpam-3781	217	1	a.v	a.v	PROPN
ejpam-3781	217	2	.	.	PROPN
ejpam-3781	217	3	ramakrishna	ramakrishna	PROPN
ejpam-3781	217	4	,	,	PUNCT
ejpam-3781	217	5	t.v.n	t.v.n	PROPN
ejpam-3781	217	6	.	.	PUNCT
ejpam-3781	218	1	prasanna	prasanna	PROPN
ejpam-3781	218	2	,	,	PUNCT
ejpam-3781	218	3	d.v	d.v	PROPN
ejpam-3781	218	4	.	.	PROPN
ejpam-3781	218	5	lakshmi	lakshmi	PROPN
ejpam-3781	218	6	/	/	SYM
ejpam-3781	218	7	eur	eur	PROPN
ejpam-3781	218	8	.	.	PUNCT
ejpam-3781	219	1	j.	j.	PROPN
ejpam-3781	219	2	pure	pure	PROPN
ejpam-3781	219	3	appl	appl	PROPN
ejpam-3781	219	4	.	.	PROPN
ejpam-3781	219	5	math	math	PROPN
ejpam-3781	219	6	,	,	PUNCT
ejpam-3781	219	7	14	14	NUM
ejpam-3781	219	8	(	(	PUNCT
ejpam-3781	219	9	1	1	NUM
ejpam-3781	219	10	)	)	PUNCT
ejpam-3781	219	11	(	(	PUNCT
ejpam-3781	219	12	2021	2021	NUM
ejpam-3781	219	13	)	)	PUNCT
ejpam-3781	219	14	,	,	PUNCT
ejpam-3781	219	15	126	126	NUM
ejpam-3781	219	16	-	-	SYM
ejpam-3781	219	17	134	134	NUM
ejpam-3781	219	18	132	132	NUM
ejpam-3781	219	19	proof	proof	NOUN
ejpam-3781	219	20	.	.	PUNCT
ejpam-3781	220	1	let	let	VERB
ejpam-3781	220	2	g	g	NOUN
ejpam-3781	220	3	=	=	PUNCT
ejpam-3781	220	4	f	f	PROPN
ejpam-3781	220	5	◦	◦	NOUN
ejpam-3781	220	6	φ	φ	NUM
ejpam-3781	220	7	.	.	PUNCT
ejpam-3781	221	1	for	for	ADP
ejpam-3781	221	2	any	any	DET
ejpam-3781	221	3	m2,m2	m2,m2	PROPN
ejpam-3781	221	4	′	′	NUM
ejpam-3781	221	5	∈m2	∈m2	NOUN
ejpam-3781	221	6	,	,	PUNCT
ejpam-3781	221	7	g(g(m2)m2	g(g(m2)m2	NOUN
ejpam-3781	221	8	′	′	NOUN
ejpam-3781	221	9	)	)	PUNCT
ejpam-3781	221	10	=	=	PUNCT
ejpam-3781	221	11	g([(f	g([(f	PROPN
ejpam-3781	221	12	◦	◦	NOUN
ejpam-3781	221	13	φ)(m2)]m2	φ)(m2)]m2	NUM
ejpam-3781	221	14	′	′	NUM
ejpam-3781	221	15	)	)	PUNCT
ejpam-3781	221	16	=	=	PUNCT
ejpam-3781	221	17	g((f(φ(m2))(m2	g((f(φ(m2))(m2	NOUN
ejpam-3781	221	18	′	′	NOUN
ejpam-3781	221	19	)	)	PUNCT
ejpam-3781	221	20	)	)	PUNCT
ejpam-3781	222	1	=	=	PRON
ejpam-3781	222	2	(	(	PUNCT
ejpam-3781	222	3	f	f	X
ejpam-3781	222	4	◦	◦	NOUN
ejpam-3781	222	5	φ)[f(φ(m2))m2	φ)[f(φ(m2))m2	PROPN
ejpam-3781	222	6	′	′	NOUN
ejpam-3781	222	7	]	]	X
ejpam-3781	223	1	=	=	PUNCT
ejpam-3781	223	2	f	f	X
ejpam-3781	224	1	[	[	X
ejpam-3781	224	2	φ[f(φ(m2))m2	φ[f(φ(m2))m2	NOUN
ejpam-3781	224	3	′	′	ADP
ejpam-3781	224	4	]	]	X
ejpam-3781	224	5	]	]	X
ejpam-3781	225	1	=	=	PUNCT
ejpam-3781	225	2	f	f	X
ejpam-3781	226	1	[	[	X
ejpam-3781	226	2	f(φ(m2))φ(m2	f(φ(m2))φ(m2	PROPN
ejpam-3781	226	3	′	′	NUM
ejpam-3781	226	4	)	)	PUNCT
ejpam-3781	226	5	]	]	PUNCT
ejpam-3781	227	1	=	=	PUNCT
ejpam-3781	227	2	f(φ(m2))f(φ(m2	f(φ(m2))f(φ(m2	NOUN
ejpam-3781	227	3	′	′	NOUN
ejpam-3781	227	4	)	)	PUNCT
ejpam-3781	227	5	)	)	PUNCT
ejpam-3781	228	1	=	=	PUNCT
ejpam-3781	228	2	g(m2)g(m2	g(m2)g(m2	PROPN
ejpam-3781	228	3	′	′	NUM
ejpam-3781	228	4	)	)	PUNCT
ejpam-3781	228	5	.	.	PUNCT
ejpam-3781	229	1	therefore	therefore	ADV
ejpam-3781	229	2	g	g	PROPN
ejpam-3781	229	3	is	be	AUX
ejpam-3781	229	4	a	a	DET
ejpam-3781	229	5	semilinear	semilinear	NOUN
ejpam-3781	229	6	map	map	NOUN
ejpam-3781	229	7	.	.	PUNCT
ejpam-3781	230	1	remark	remark	PROPN
ejpam-3781	230	2	3	3	NUM
ejpam-3781	230	3	.	.	PUNCT
ejpam-3781	230	4	suppose	suppose	VERB
ejpam-3781	230	5	a	a	DET
ejpam-3781	230	6	modified	modify	VERB
ejpam-3781	230	7	near	near	ADP
ejpam-3781	230	8	module	module	NOUN
ejpam-3781	230	9	(	(	PUNCT
ejpam-3781	230	10	m,+	m,+	PROPN
ejpam-3781	230	11	,	,	PUNCT
ejpam-3781	230	12	·	·	PUNCT
ejpam-3781	230	13	)	)	PUNCT
ejpam-3781	230	14	over	over	ADP
ejpam-3781	230	15	a	a	DET
ejpam-3781	230	16	near	near	ADJ
ejpam-3781	230	17	ring	ring	NOUN
ejpam-3781	230	18	(	(	PUNCT
ejpam-3781	230	19	n,+	n,+	NUM
ejpam-3781	230	20	,	,	PUNCT
ejpam-3781	230	21	·	·	PUNCT
ejpam-3781	230	22	)	)	PUNCT
ejpam-3781	230	23	is	be	AUX
ejpam-3781	230	24	made	make	VERB
ejpam-3781	230	25	into	into	ADP
ejpam-3781	230	26	a	a	DET
ejpam-3781	230	27	near	near	ADJ
ejpam-3781	230	28	ring	ring	NOUN
ejpam-3781	230	29	(	(	PUNCT
ejpam-3781	230	30	m,+	m,+	PROPN
ejpam-3781	230	31	,	,	PUNCT
ejpam-3781	230	32	∗	∗	NOUN
ejpam-3781	230	33	)	)	PUNCT
ejpam-3781	230	34	with	with	ADP
ejpam-3781	230	35	the	the	DET
ejpam-3781	230	36	help	help	NOUN
ejpam-3781	230	37	of	of	ADP
ejpam-3781	230	38	a	a	DET
ejpam-3781	230	39	semilinear	semilinear	NOUN
ejpam-3781	230	40	map	map	NOUN
ejpam-3781	231	1	f	f	PROPN
ejpam-3781	231	2	.	.	PUNCT
ejpam-3781	232	1	then	then	ADV
ejpam-3781	232	2	we	we	PRON
ejpam-3781	232	3	know	know	VERB
ejpam-3781	232	4	that	that	SCONJ
ejpam-3781	232	5	mk	mk	PROPN
ejpam-3781	232	6	,	,	PUNCT
ejpam-3781	232	7	the	the	DET
ejpam-3781	232	8	k	k	ADJ
ejpam-3781	232	9	-	-	ADJ
ejpam-3781	232	10	fold	fold	ADJ
ejpam-3781	232	11	product	product	NOUN
ejpam-3781	232	12	of	of	ADP
ejpam-3781	232	13	(	(	PUNCT
ejpam-3781	232	14	m,+	m,+	INTJ
ejpam-3781	232	15	,	,	PUNCT
ejpam-3781	232	16	∗	∗	NOUN
ejpam-3781	232	17	)	)	PUNCT
ejpam-3781	232	18	is	be	AUX
ejpam-3781	232	19	also	also	ADV
ejpam-3781	232	20	a	a	DET
ejpam-3781	232	21	near	near	ADJ
ejpam-3781	232	22	ring	ring	NOUN
ejpam-3781	232	23	.	.	PUNCT
ejpam-3781	233	1	it	it	PRON
ejpam-3781	233	2	may	may	AUX
ejpam-3781	233	3	be	be	AUX
ejpam-3781	233	4	hoped	hope	VERB
ejpam-3781	233	5	that	that	SCONJ
ejpam-3781	233	6	the	the	DET
ejpam-3781	233	7	near	near	ADJ
ejpam-3781	233	8	ring	ring	NOUN
ejpam-3781	233	9	module	module	NOUN
ejpam-3781	233	10	(	(	PUNCT
ejpam-3781	233	11	mk,⊕	mk,⊕	PROPN
ejpam-3781	233	12	,	,	PUNCT
ejpam-3781	233	13	·	·	PUNCT
ejpam-3781	233	14	)	)	PUNCT
ejpam-3781	233	15	can	can	AUX
ejpam-3781	233	16	be	be	AUX
ejpam-3781	233	17	made	make	VERB
ejpam-3781	233	18	into	into	ADP
ejpam-3781	233	19	the	the	DET
ejpam-3781	233	20	near	near	ADJ
ejpam-3781	233	21	ring	ring	NOUN
ejpam-3781	233	22	(	(	PUNCT
ejpam-3781	233	23	mk,⊕,⊗	mk,⊕,⊗	NOUN
ejpam-3781	233	24	)	)	PUNCT
ejpam-3781	233	25	directly	directly	ADV
ejpam-3781	233	26	by	by	ADP
ejpam-3781	233	27	employing	employ	VERB
ejpam-3781	233	28	a	a	DET
ejpam-3781	233	29	suitable	suitable	ADJ
ejpam-3781	233	30	semilinear	semilinear	NOUN
ejpam-3781	233	31	map	map	NOUN
ejpam-3781	233	32	from	from	ADP
ejpam-3781	233	33	mk	mk	NOUN
ejpam-3781	233	34	into	into	ADP
ejpam-3781	233	35	n	n	PROPN
ejpam-3781	233	36	.	.	PUNCT
ejpam-3781	234	1	the	the	DET
ejpam-3781	234	2	following	following	ADJ
ejpam-3781	234	3	example	example	NOUN
ejpam-3781	234	4	warns	warn	VERB
ejpam-3781	234	5	that	that	SCONJ
ejpam-3781	234	6	not	not	PART
ejpam-3781	234	7	every	every	PRON
ejpam-3781	234	8	modified	modify	VERB
ejpam-3781	234	9	near	near	ADP
ejpam-3781	234	10	module	module	NOUN
ejpam-3781	234	11	comes	come	VERB
ejpam-3781	234	12	through	through	ADP
ejpam-3781	234	13	a	a	DET
ejpam-3781	234	14	semilinear	semilinear	NOUN
ejpam-3781	234	15	map	map	NOUN
ejpam-3781	234	16	.	.	PUNCT
ejpam-3781	235	1	as	as	ADP
ejpam-3781	235	2	an	an	DET
ejpam-3781	235	3	illustration	illustration	NOUN
ejpam-3781	235	4	we	we	PRON
ejpam-3781	235	5	present	present	VERB
ejpam-3781	235	6	the	the	DET
ejpam-3781	235	7	following	following	NOUN
ejpam-3781	235	8	:	:	PUNCT
ejpam-3781	235	9	example	example	NOUN
ejpam-3781	235	10	10	10	NUM
ejpam-3781	235	11	.	.	PUNCT
ejpam-3781	236	1	define	define	VERB
ejpam-3781	236	2	x	x	DET
ejpam-3781	236	3	·	·	PUNCT
ejpam-3781	236	4	y	y	NOUN
ejpam-3781	236	5	=	=	PUNCT
ejpam-3781	236	6	2xy	2xy	NOUN
ejpam-3781	236	7	for	for	ADP
ejpam-3781	236	8	all	all	DET
ejpam-3781	236	9	x	x	NOUN
ejpam-3781	236	10	,	,	PUNCT
ejpam-3781	236	11	y	y	PROPN
ejpam-3781	236	12	∈	∈	PROPN
ejpam-3781	236	13	r.	r.	PROPN
ejpam-3781	236	14	then	then	ADV
ejpam-3781	236	15	(	(	PUNCT
ejpam-3781	236	16	r,+	r,+	NUM
ejpam-3781	236	17	,	,	PUNCT
ejpam-3781	236	18	·	·	PUNCT
ejpam-3781	236	19	)	)	PUNCT
ejpam-3781	236	20	is	be	AUX
ejpam-3781	236	21	a	a	DET
ejpam-3781	236	22	modified	modify	VERB
ejpam-3781	236	23	near	near	ADP
ejpam-3781	236	24	module	module	NOUN
ejpam-3781	236	25	over	over	ADP
ejpam-3781	236	26	r.	r.	PROPN
ejpam-3781	236	27	define	define	VERB
ejpam-3781	236	28	f	f	PROPN
ejpam-3781	236	29	:	:	PUNCT
ejpam-3781	236	30	r→	r→	PROPN
ejpam-3781	236	31	r	r	NOUN
ejpam-3781	236	32	by	by	ADP
ejpam-3781	236	33	f(m	f(m	PROPN
ejpam-3781	236	34	)	)	PUNCT
ejpam-3781	236	35	=	=	PUNCT
ejpam-3781	237	1	2	2	NUM
ejpam-3781	237	2	m	m	VERB
ejpam-3781	237	3	for	for	ADP
ejpam-3781	237	4	all	all	DET
ejpam-3781	237	5	m	m	PROPN
ejpam-3781	237	6	∈	∈	PROPN
ejpam-3781	237	7	r.	r.	NOUN
ejpam-3781	237	8	then	then	ADV
ejpam-3781	237	9	f(f(a	f(f(a	NOUN
ejpam-3781	237	10	)	)	PUNCT
ejpam-3781	237	11	·	·	PUNCT
ejpam-3781	238	1	b	b	X
ejpam-3781	238	2	)	)	PUNCT
ejpam-3781	238	3	=	=	SYM
ejpam-3781	238	4	f(a	f(a	PROPN
ejpam-3781	238	5	)	)	PUNCT
ejpam-3781	238	6	·	·	PUNCT
ejpam-3781	239	1	f(b	f(b	X
ejpam-3781	239	2	)	)	PUNCT
ejpam-3781	239	3	=	=	SYM
ejpam-3781	239	4	8ab	8ab	NOUN
ejpam-3781	239	5	.	.	PUNCT
ejpam-3781	240	1	so	so	ADV
ejpam-3781	240	2	that	that	SCONJ
ejpam-3781	240	3	f	f	PROPN
ejpam-3781	240	4	is	be	AUX
ejpam-3781	240	5	semilinear	semilinear	ADJ
ejpam-3781	240	6	.	.	PUNCT
ejpam-3781	241	1	now	now	ADV
ejpam-3781	241	2	m1	m1	PROPN
ejpam-3781	241	3	∗m2	∗m2	PROPN
ejpam-3781	241	4	=	=	PROPN
ejpam-3781	241	5	f(m2	f(m2	NOUN
ejpam-3781	241	6	)	)	PUNCT
ejpam-3781	241	7	·	·	PUNCT
ejpam-3781	241	8	m1	m1	NOUN
ejpam-3781	241	9	=	=	SYM
ejpam-3781	241	10	2f(m2)m1	2f(m2)m1	NUM
ejpam-3781	241	11	=	=	SYM
ejpam-3781	241	12	2(2m2)m1	2(2m2)m1	NUM
ejpam-3781	241	13	=	=	SYM
ejpam-3781	241	14	4m2m1	4m2m1	NOUN
ejpam-3781	241	15	.	.	PUNCT
ejpam-3781	242	1	consider	consider	VERB
ejpam-3781	242	2	(	(	PUNCT
ejpam-3781	242	3	r2,⊕,⊗	r2,⊕,⊗	NOUN
ejpam-3781	242	4	)	)	PUNCT
ejpam-3781	242	5	,	,	PUNCT
ejpam-3781	242	6	the	the	DET
ejpam-3781	242	7	product	product	NOUN
ejpam-3781	242	8	of	of	ADP
ejpam-3781	242	9	the	the	DET
ejpam-3781	242	10	near	near	ADJ
ejpam-3781	242	11	ring	ring	NOUN
ejpam-3781	242	12	(	(	PUNCT
ejpam-3781	242	13	r,+	r,+	NUM
ejpam-3781	242	14	,	,	PUNCT
ejpam-3781	242	15	∗	∗	NOUN
ejpam-3781	242	16	)	)	PUNCT
ejpam-3781	242	17	with	with	ADP
ejpam-3781	242	18	itself	itself	PRON
ejpam-3781	242	19	.	.	PUNCT
ejpam-3781	243	1	suppose	suppose	VERB
ejpam-3781	243	2	if	if	SCONJ
ejpam-3781	243	3	possible	possible	ADJ
ejpam-3781	243	4	there	there	PRON
ejpam-3781	243	5	is	be	VERB
ejpam-3781	243	6	a	a	DET
ejpam-3781	243	7	semilinear	semilinear	ADJ
ejpam-3781	243	8	map	map	NOUN
ejpam-3781	243	9	g	g	PROPN
ejpam-3781	243	10	:	:	PUNCT
ejpam-3781	243	11	r2	r2	PROPN
ejpam-3781	243	12	→	→	PUNCT
ejpam-3781	243	13	r	r	NOUN
ejpam-3781	243	14	such	such	ADJ
ejpam-3781	243	15	that	that	SCONJ
ejpam-3781	243	16	‘	'	PUNCT
ejpam-3781	243	17	⊗	⊗	NOUN
ejpam-3781	243	18	’	'	PUNCT
ejpam-3781	243	19	is	be	AUX
ejpam-3781	243	20	induced	induce	VERB
ejpam-3781	243	21	by	by	ADP
ejpam-3781	243	22	g.	g.	PROPN
ejpam-3781	243	23	now	now	ADV
ejpam-3781	243	24	(	(	PUNCT
ejpam-3781	243	25	m1	m1	PROPN
ejpam-3781	243	26	·	·	SYM
ejpam-3781	243	27	m3,m2	m3,m2	PROPN
ejpam-3781	243	28	·	·	SYM
ejpam-3781	243	29	m4	m4	PROPN
ejpam-3781	243	30	)	)	PUNCT
ejpam-3781	243	31	=	=	PUNCT
ejpam-3781	243	32	(	(	PUNCT
ejpam-3781	243	33	m1,m2)⊗	m1,m2)⊗	X
ejpam-3781	243	34	(	(	PUNCT
ejpam-3781	243	35	m3,m4	m3,m4	PROPN
ejpam-3781	243	36	)	)	PUNCT
ejpam-3781	243	37	=	=	SYM
ejpam-3781	243	38	g(m3,m4	g(m3,m4	PROPN
ejpam-3781	243	39	)	)	PUNCT
ejpam-3781	243	40	·	·	PUNCT
ejpam-3781	243	41	(	(	PUNCT
ejpam-3781	243	42	m1,m2	m1,m2	PROPN
ejpam-3781	243	43	)	)	PUNCT
ejpam-3781	243	44	=	=	PUNCT
ejpam-3781	243	45	(	(	PUNCT
ejpam-3781	243	46	α	α	NOUN
ejpam-3781	243	47	·	·	SYM
ejpam-3781	243	48	m1	m1	NOUN
ejpam-3781	243	49	,	,	PUNCT
ejpam-3781	243	50	α	α	PROPN
ejpam-3781	243	51	·	·	SYM
ejpam-3781	243	52	m2	m2	PROPN
ejpam-3781	243	53	)	)	PUNCT
ejpam-3781	243	54	where	where	SCONJ
ejpam-3781	243	55	α	α	NOUN
ejpam-3781	243	56	=	=	SYM
ejpam-3781	243	57	g(m3,m4	g(m3,m4	PROPN
ejpam-3781	243	58	)	)	PUNCT
ejpam-3781	243	59	⇒	⇒	PROPN
ejpam-3781	243	60	m1	m1	PROPN
ejpam-3781	243	61	·	·	PUNCT
ejpam-3781	243	62	m3	m3	PROPN
ejpam-3781	243	63	=	=	PUNCT
ejpam-3781	243	64	α	α	PROPN
ejpam-3781	243	65	·	·	PUNCT
ejpam-3781	243	66	m1	m1	PROPN
ejpam-3781	243	67	and	and	CCONJ
ejpam-3781	243	68	m2	m2	PROPN
ejpam-3781	243	69	·	·	PUNCT
ejpam-3781	243	70	m4	m4	PROPN
ejpam-3781	243	71	=	=	PUNCT
ejpam-3781	243	72	α	α	PROPN
ejpam-3781	243	73	·	·	PUNCT
ejpam-3781	243	74	m2	m2	PROPN
ejpam-3781	243	75	⇒	⇒	PROPN
ejpam-3781	243	76	2m1m3	2m1m3	VERB
ejpam-3781	244	1	=	=	SYM
ejpam-3781	244	2	2αm1	2αm1	NUM
ejpam-3781	244	3	and	and	CCONJ
ejpam-3781	244	4	2m2m4	2m2m4	NUM
ejpam-3781	244	5	=	=	SYM
ejpam-3781	244	6	2αm2	2αm2	NUM
ejpam-3781	244	7	for	for	ADP
ejpam-3781	244	8	all	all	DET
ejpam-3781	244	9	m1,m2,m3,m4	m1,m2,m3,m4	PROPN
ejpam-3781	244	10	∈m	∈m	NOUN
ejpam-3781	244	11	.	.	PUNCT
ejpam-3781	245	1	taking	take	VERB
ejpam-3781	245	2	m1	m1	NOUN
ejpam-3781	245	3	=	=	SYM
ejpam-3781	245	4	m3	m3	PROPN
ejpam-3781	245	5	=	=	SYM
ejpam-3781	245	6	1,m2	1,m2	NUM
ejpam-3781	245	7	=	=	SYM
ejpam-3781	245	8	m4	m4	PROPN
ejpam-3781	245	9	=	=	SYM
ejpam-3781	245	10	2	2	NUM
ejpam-3781	245	11	,	,	PUNCT
ejpam-3781	245	12	we	we	PRON
ejpam-3781	245	13	get	get	VERB
ejpam-3781	245	14	2	2	NUM
ejpam-3781	245	15	=	=	NOUN
ejpam-3781	245	16	2α	2α	NOUN
ejpam-3781	245	17	and	and	CCONJ
ejpam-3781	245	18	8	8	NUM
ejpam-3781	245	19	=	=	SYM
ejpam-3781	245	20	4α	4α	NOUN
ejpam-3781	245	21	⇒	⇒	NOUN
ejpam-3781	245	22	α	α	X
ejpam-3781	245	23	=	=	SYM
ejpam-3781	245	24	1	1	NUM
ejpam-3781	245	25	and	and	CCONJ
ejpam-3781	245	26	α	α	NOUN
ejpam-3781	245	27	=	=	SYM
ejpam-3781	245	28	2	2	NUM
ejpam-3781	245	29	,	,	PUNCT
ejpam-3781	245	30	which	which	PRON
ejpam-3781	245	31	is	be	AUX
ejpam-3781	245	32	a	a	DET
ejpam-3781	245	33	contradiction	contradiction	NOUN
ejpam-3781	245	34	.	.	PUNCT
ejpam-3781	246	1	theorem	theorem	ADJ
ejpam-3781	246	2	7	7	NUM
ejpam-3781	246	3	.	.	PUNCT
ejpam-3781	247	1	let	let	VERB
ejpam-3781	247	2	m	m	PRON
ejpam-3781	247	3	be	be	AUX
ejpam-3781	247	4	a	a	DET
ejpam-3781	247	5	modified	modify	VERB
ejpam-3781	247	6	near	near	ADP
ejpam-3781	247	7	module	module	NOUN
ejpam-3781	247	8	over	over	ADP
ejpam-3781	247	9	(	(	PUNCT
ejpam-3781	247	10	r,+	r,+	NUM
ejpam-3781	247	11	)	)	PUNCT
ejpam-3781	247	12	and	and	CCONJ
ejpam-3781	247	13	f	f	X
ejpam-3781	247	14	:	:	PUNCT
ejpam-3781	247	15	m	m	AUX
ejpam-3781	247	16	→	→	SYM
ejpam-3781	247	17	r	r	AUX
ejpam-3781	247	18	be	be	AUX
ejpam-3781	247	19	a	a	DET
ejpam-3781	247	20	semilinear	semilinear	NOUN
ejpam-3781	247	21	map	map	NOUN
ejpam-3781	247	22	.	.	PUNCT
ejpam-3781	248	1	(	(	PUNCT
ejpam-3781	248	2	i	i	NOUN
ejpam-3781	248	3	)	)	PUNCT
ejpam-3781	248	4	if	if	SCONJ
ejpam-3781	248	5	f	f	PROPN
ejpam-3781	248	6	is	be	AUX
ejpam-3781	248	7	one	one	NUM
ejpam-3781	248	8	-	-	PUNCT
ejpam-3781	248	9	one	one	NUM
ejpam-3781	248	10	,	,	PUNCT
ejpam-3781	248	11	then	then	ADV
ejpam-3781	248	12	(	(	PUNCT
ejpam-3781	248	13	mf	mf	X
ejpam-3781	248	14	,	,	PUNCT
ejpam-3781	248	15	+	+	PROPN
ejpam-3781	248	16	,	,	PUNCT
ejpam-3781	248	17	∗	∗	NOUN
ejpam-3781	248	18	)	)	PUNCT
ejpam-3781	248	19	is	be	AUX
ejpam-3781	248	20	commutative	commutative	ADJ
ejpam-3781	248	21	.	.	PUNCT
ejpam-3781	249	1	(	(	PUNCT
ejpam-3781	249	2	ii	ii	NOUN
ejpam-3781	249	3	)	)	PUNCT
ejpam-3781	249	4	suppose	suppose	VERB
ejpam-3781	249	5	m	m	VERB
ejpam-3781	249	6	=	=	SYM
ejpam-3781	249	7	(	(	PUNCT
ejpam-3781	249	8	rk,+	rk,+	PROPN
ejpam-3781	249	9	)	)	PUNCT
ejpam-3781	249	10	.	.	PUNCT
ejpam-3781	250	1	then	then	ADV
ejpam-3781	250	2	(	(	PUNCT
ejpam-3781	250	3	mf	mf	X
ejpam-3781	250	4	,	,	PUNCT
ejpam-3781	250	5	+	+	PROPN
ejpam-3781	250	6	,	,	PUNCT
ejpam-3781	250	7	∗	∗	NOUN
ejpam-3781	250	8	)	)	PUNCT
ejpam-3781	250	9	is	be	AUX
ejpam-3781	250	10	commutative	commutative	ADJ
ejpam-3781	250	11	if	if	SCONJ
ejpam-3781	250	12	and	and	CCONJ
ejpam-3781	250	13	only	only	ADV
ejpam-3781	250	14	if	if	SCONJ
ejpam-3781	250	15	either	either	PRON
ejpam-3781	250	16	m	m	VERB
ejpam-3781	250	17	=	=	SYM
ejpam-3781	250	18	{	{	PUNCT
ejpam-3781	250	19	0	0	NUM
ejpam-3781	250	20	}	}	PUNCT
ejpam-3781	250	21	or	or	CCONJ
ejpam-3781	250	22	(	(	PUNCT
ejpam-3781	250	23	mf	mf	X
ejpam-3781	250	24	,	,	PUNCT
ejpam-3781	250	25	+	+	PROPN
ejpam-3781	250	26	,	,	PUNCT
ejpam-3781	250	27	∗	∗	NOUN
ejpam-3781	250	28	)	)	PUNCT
ejpam-3781	250	29	'	'	PUNCT
ejpam-3781	250	30	(	(	PUNCT
ejpam-3781	250	31	r,+	r,+	NUM
ejpam-3781	250	32	,	,	PUNCT
ejpam-3781	250	33	·	·	PUNCT
ejpam-3781	250	34	)	)	PUNCT
ejpam-3781	250	35	,	,	PUNCT
ejpam-3781	250	36	where	where	SCONJ
ejpam-3781	250	37	mf	mf	NOUN
ejpam-3781	250	38	is	be	AUX
ejpam-3781	250	39	a	a	DET
ejpam-3781	250	40	near	near	ADJ
ejpam-3781	250	41	ring	ring	NOUN
ejpam-3781	250	42	induced	induce	VERB
ejpam-3781	250	43	by	by	ADP
ejpam-3781	250	44	the	the	DET
ejpam-3781	250	45	semilinear	semilinear	PROPN
ejpam-3781	250	46	map	map	NOUN
ejpam-3781	251	1	f	f	PROPN
ejpam-3781	251	2	.	.	PUNCT
ejpam-3781	252	1	a.v	a.v	PROPN
ejpam-3781	252	2	.	.	PROPN
ejpam-3781	252	3	ramakrishna	ramakrishna	PROPN
ejpam-3781	252	4	,	,	PUNCT
ejpam-3781	252	5	t.v.n	t.v.n	PROPN
ejpam-3781	252	6	.	.	PUNCT
ejpam-3781	253	1	prasanna	prasanna	PROPN
ejpam-3781	253	2	,	,	PUNCT
ejpam-3781	253	3	d.v	d.v	PROPN
ejpam-3781	253	4	.	.	PROPN
ejpam-3781	253	5	lakshmi	lakshmi	PROPN
ejpam-3781	253	6	/	/	SYM
ejpam-3781	253	7	eur	eur	PROPN
ejpam-3781	253	8	.	.	PUNCT
ejpam-3781	254	1	j.	j.	PROPN
ejpam-3781	254	2	pure	pure	PROPN
ejpam-3781	254	3	appl	appl	PROPN
ejpam-3781	254	4	.	.	PROPN
ejpam-3781	254	5	math	math	PROPN
ejpam-3781	254	6	,	,	PUNCT
ejpam-3781	254	7	14	14	NUM
ejpam-3781	254	8	(	(	PUNCT
ejpam-3781	254	9	1	1	NUM
ejpam-3781	254	10	)	)	PUNCT
ejpam-3781	254	11	(	(	PUNCT
ejpam-3781	254	12	2021	2021	NUM
ejpam-3781	254	13	)	)	PUNCT
ejpam-3781	254	14	,	,	PUNCT
ejpam-3781	254	15	126	126	NUM
ejpam-3781	254	16	-	-	SYM
ejpam-3781	254	17	134	134	NUM
ejpam-3781	254	18	133	133	NUM
ejpam-3781	254	19	proof	proof	NOUN
ejpam-3781	254	20	.	.	PUNCT
ejpam-3781	255	1	(	(	PUNCT
ejpam-3781	255	2	1	1	X
ejpam-3781	255	3	)	)	PUNCT
ejpam-3781	255	4	for	for	ADP
ejpam-3781	255	5	any	any	DET
ejpam-3781	255	6	m1,m2	m1,m2	PROPN
ejpam-3781	255	7	∈m	∈m	NOUN
ejpam-3781	255	8	,	,	PUNCT
ejpam-3781	255	9	m1	m1	PROPN
ejpam-3781	255	10	∗m2	∗m2	PROPN
ejpam-3781	255	11	=	=	PUNCT
ejpam-3781	255	12	f(m2)m1	f(m2)m1	PROPN
ejpam-3781	255	13	and	and	CCONJ
ejpam-3781	255	14	m2	m2	PROPN
ejpam-3781	255	15	∗m1	∗m1	PUNCT
ejpam-3781	255	16	=	=	PUNCT
ejpam-3781	255	17	f(m1)m2	f(m1)m2	NOUN
ejpam-3781	255	18	.	.	PUNCT
ejpam-3781	256	1	now	now	ADV
ejpam-3781	256	2	f(m1	f(m1	VERB
ejpam-3781	256	3	∗m2	∗m2	NOUN
ejpam-3781	256	4	)	)	PUNCT
ejpam-3781	256	5	=	=	SYM
ejpam-3781	256	6	f(f(m2)m1	f(f(m2)m1	NOUN
ejpam-3781	256	7	)	)	PUNCT
ejpam-3781	256	8	=	=	SYM
ejpam-3781	256	9	f(m2)f(m1	f(m2)f(m1	NOUN
ejpam-3781	256	10	)	)	PUNCT
ejpam-3781	256	11	.	.	PUNCT
ejpam-3781	257	1	also	also	ADV
ejpam-3781	257	2	f(m2	f(m2	PROPN
ejpam-3781	257	3	∗m1	∗m1	X
ejpam-3781	257	4	)	)	PUNCT
ejpam-3781	257	5	=	=	SYM
ejpam-3781	257	6	f(f(m1)m2	f(f(m1)m2	ADJ
ejpam-3781	257	7	)	)	PUNCT
ejpam-3781	257	8	=	=	SYM
ejpam-3781	257	9	f(m1)f(m2	f(m1)f(m2	PROPN
ejpam-3781	257	10	)	)	PUNCT
ejpam-3781	257	11	.	.	PUNCT
ejpam-3781	258	1	since	since	SCONJ
ejpam-3781	258	2	(	(	PUNCT
ejpam-3781	258	3	r	r	NOUN
ejpam-3781	258	4	,	,	PUNCT
ejpam-3781	258	5	·	·	PUNCT
ejpam-3781	258	6	)	)	PUNCT
ejpam-3781	258	7	is	be	AUX
ejpam-3781	258	8	commutative	commutative	ADJ
ejpam-3781	258	9	,	,	PUNCT
ejpam-3781	258	10	we	we	PRON
ejpam-3781	258	11	have	have	VERB
ejpam-3781	258	12	f(m1	f(m1	NOUN
ejpam-3781	258	13	∗m2	∗m2	NOUN
ejpam-3781	258	14	)	)	PUNCT
ejpam-3781	258	15	=	=	PUNCT
ejpam-3781	258	16	f(m2	f(m2	NOUN
ejpam-3781	258	17	∗m1	∗m1	PROPN
ejpam-3781	258	18	)	)	PUNCT
ejpam-3781	258	19	.	.	PUNCT
ejpam-3781	259	1	since	since	SCONJ
ejpam-3781	259	2	f	f	PROPN
ejpam-3781	259	3	is	be	AUX
ejpam-3781	259	4	one	one	NUM
ejpam-3781	259	5	-	-	PUNCT
ejpam-3781	259	6	one	one	NUM
ejpam-3781	259	7	,	,	PUNCT
ejpam-3781	259	8	we	we	PRON
ejpam-3781	259	9	have	have	VERB
ejpam-3781	259	10	m1	m1	PROPN
ejpam-3781	259	11	∗m2	∗m2	PROPN
ejpam-3781	259	12	=	=	PROPN
ejpam-3781	259	13	m2	m2	PROPN
ejpam-3781	259	14	∗m1	∗m1	PROPN
ejpam-3781	259	15	.	.	PUNCT
ejpam-3781	260	1	so	so	ADV
ejpam-3781	260	2	‘	'	PUNCT
ejpam-3781	260	3	*	*	PUNCT
ejpam-3781	260	4	’	'	PUNCT
ejpam-3781	260	5	is	be	AUX
ejpam-3781	260	6	commutative	commutative	ADJ
ejpam-3781	260	7	on	on	ADP
ejpam-3781	260	8	m	m	PRON
ejpam-3781	260	9	and	and	CCONJ
ejpam-3781	260	10	hence	hence	ADV
ejpam-3781	260	11	(	(	PUNCT
ejpam-3781	260	12	mf	mf	X
ejpam-3781	260	13	,	,	PUNCT
ejpam-3781	260	14	+	+	PROPN
ejpam-3781	260	15	,	,	PUNCT
ejpam-3781	260	16	∗	∗	NOUN
ejpam-3781	260	17	)	)	PUNCT
ejpam-3781	260	18	is	be	AUX
ejpam-3781	260	19	commutative	commutative	ADJ
ejpam-3781	260	20	.	.	PUNCT
ejpam-3781	261	1	(	(	PUNCT
ejpam-3781	261	2	2	2	X
ejpam-3781	261	3	)	)	PUNCT
ejpam-3781	261	4	suppose	suppose	VERB
ejpam-3781	261	5	(	(	PUNCT
ejpam-3781	261	6	mf	mf	X
ejpam-3781	261	7	,	,	PUNCT
ejpam-3781	261	8	+	+	PROPN
ejpam-3781	261	9	,	,	PUNCT
ejpam-3781	261	10	∗	∗	NOUN
ejpam-3781	261	11	)	)	PUNCT
ejpam-3781	261	12	is	be	AUX
ejpam-3781	261	13	commutative	commutative	ADJ
ejpam-3781	261	14	.	.	PUNCT
ejpam-3781	262	1	then	then	ADV
ejpam-3781	262	2	m1	m1	PROPN
ejpam-3781	262	3	∗m2	∗m2	PROPN
ejpam-3781	262	4	=	=	PROPN
ejpam-3781	262	5	m2	m2	PROPN
ejpam-3781	262	6	∗m1	∗m1	SYM
ejpam-3781	262	7	⇒f(m2)m1	⇒f(m2)m1	NUM
ejpam-3781	262	8	=	=	SYM
ejpam-3781	262	9	f(m1)m2	f(m1)m2	NOUN
ejpam-3781	262	10	⇒	⇒	VERB
ejpam-3781	262	11	the	the	DET
ejpam-3781	262	12	vectors	vector	NOUN
ejpam-3781	262	13	m1	m1	PROPN
ejpam-3781	262	14	and	and	CCONJ
ejpam-3781	262	15	m2	m2	PROPN
ejpam-3781	262	16	are	be	AUX
ejpam-3781	262	17	parallel	parallel	ADJ
ejpam-3781	263	1	⇒k	⇒k	NOUN
ejpam-3781	263	2	=	=	SYM
ejpam-3781	263	3	0	0	NUM
ejpam-3781	263	4	or	or	CCONJ
ejpam-3781	263	5	k	k	X
ejpam-3781	263	6	=	=	SYM
ejpam-3781	263	7	1	1	X
ejpam-3781	263	8	.	.	PUNCT
ejpam-3781	263	9	when	when	SCONJ
ejpam-3781	263	10	k=1	k=1	NOUN
ejpam-3781	263	11	:	:	PUNCT
ejpam-3781	263	12	now	now	ADV
ejpam-3781	263	13	m1	m1	PROPN
ejpam-3781	263	14	∗	∗	NOUN
ejpam-3781	263	15	m2	m2	PROPN
ejpam-3781	263	16	=	=	PUNCT
ejpam-3781	263	17	f(m2)m1	f(m2)m1	PROPN
ejpam-3781	263	18	and	and	CCONJ
ejpam-3781	263	19	m2	m2	PROPN
ejpam-3781	263	20	∗	∗	PROPN
ejpam-3781	263	21	m1	m1	PROPN
ejpam-3781	263	22	=	=	SYM
ejpam-3781	263	23	f(m1)m2	f(m1)m2	NOUN
ejpam-3781	263	24	⇒	⇒	NOUN
ejpam-3781	263	25	f(m2)m1	f(m2)m1	NOUN
ejpam-3781	263	26	=	=	SYM
ejpam-3781	263	27	f(m1)m2	f(m1)m2	NOUN
ejpam-3781	263	28	for	for	ADP
ejpam-3781	263	29	all	all	DET
ejpam-3781	263	30	m1,m2	m1,m2	PROPN
ejpam-3781	263	31	∈m	∈m	NOUN
ejpam-3781	263	32	.	.	PUNCT
ejpam-3781	264	1	this	this	DET
ejpam-3781	264	2	equality	equality	NOUN
ejpam-3781	264	3	is	be	AUX
ejpam-3781	264	4	true	true	ADJ
ejpam-3781	264	5	for	for	ADP
ejpam-3781	264	6	m1	m1	NOUN
ejpam-3781	264	7	=	=	SYM
ejpam-3781	264	8	1	1	NUM
ejpam-3781	264	9	,	,	PUNCT
ejpam-3781	264	10	we	we	PRON
ejpam-3781	264	11	get	get	VERB
ejpam-3781	264	12	f(m2	f(m2	NOUN
ejpam-3781	264	13	)	)	PUNCT
ejpam-3781	264	14	=	=	SYM
ejpam-3781	264	15	f(1)m2	f(1)m2	NOUN
ejpam-3781	264	16	.	.	PUNCT
ejpam-3781	265	1	put	put	VERB
ejpam-3781	265	2	f(1	f(1	PROPN
ejpam-3781	265	3	)	)	PUNCT
ejpam-3781	266	1	=	=	SYM
ejpam-3781	266	2	λ⇒	λ⇒	NOUN
ejpam-3781	266	3	f(m2	f(m2	NOUN
ejpam-3781	266	4	)	)	PUNCT
ejpam-3781	267	1	=	=	SYM
ejpam-3781	267	2	λm2	λm2	NOUN
ejpam-3781	267	3	for	for	ADP
ejpam-3781	267	4	some	some	DET
ejpam-3781	267	5	constant	constant	ADJ
ejpam-3781	267	6	.	.	PUNCT
ejpam-3781	268	1	therefore	therefore	ADV
ejpam-3781	268	2	f	f	PROPN
ejpam-3781	268	3	is	be	AUX
ejpam-3781	268	4	linear	linear	ADJ
ejpam-3781	268	5	.	.	PUNCT
ejpam-3781	269	1	now	now	ADV
ejpam-3781	269	2	m1	m1	PROPN
ejpam-3781	269	3	∗	∗	NOUN
ejpam-3781	269	4	(	(	PUNCT
ejpam-3781	269	5	m2	m2	PROPN
ejpam-3781	269	6	+	+	PROPN
ejpam-3781	269	7	m3	m3	PROPN
ejpam-3781	269	8	)	)	PUNCT
ejpam-3781	269	9	=	=	PUNCT
ejpam-3781	269	10	f(m2	f(m2	X
ejpam-3781	270	1	+	+	NOUN
ejpam-3781	270	2	m3)m1	m3)m1	NOUN
ejpam-3781	270	3	=	=	SYM
ejpam-3781	270	4	[	[	X
ejpam-3781	270	5	f(m2	f(m2	NOUN
ejpam-3781	270	6	)	)	PUNCT
ejpam-3781	270	7	+	+	NUM
ejpam-3781	270	8	f(m3)]m1	f(m3)]m1	NOUN
ejpam-3781	270	9	=	=	SYM
ejpam-3781	270	10	f(m2)m1	f(m2)m1	NOUN
ejpam-3781	270	11	+	+	CCONJ
ejpam-3781	270	12	f(m3)m1	f(m3)m1	NOUN
ejpam-3781	270	13	=	=	SYM
ejpam-3781	270	14	m1	m1	PROPN
ejpam-3781	270	15	∗m2	∗m2	PROPN
ejpam-3781	270	16	+	+	PROPN
ejpam-3781	270	17	m1	m1	PROPN
ejpam-3781	270	18	∗m3	∗m3	PROPN
ejpam-3781	270	19	.	.	PUNCT
ejpam-3781	271	1	therefore	therefore	ADV
ejpam-3781	271	2	(	(	PUNCT
ejpam-3781	271	3	mf	mf	X
ejpam-3781	271	4	,	,	PUNCT
ejpam-3781	271	5	+	+	PROPN
ejpam-3781	271	6	,	,	PUNCT
ejpam-3781	271	7	∗	∗	NOUN
ejpam-3781	271	8	)	)	PUNCT
ejpam-3781	271	9	is	be	AUX
ejpam-3781	271	10	a	a	DET
ejpam-3781	271	11	commutative	commutative	ADJ
ejpam-3781	271	12	ring	ring	NOUN
ejpam-3781	271	13	.	.	PUNCT
ejpam-3781	272	1	let	let	VERB
ejpam-3781	272	2	0	0	NUM
ejpam-3781	273	1	6=	6=	ADP
ejpam-3781	273	2	m	m	NOUN
ejpam-3781	273	3	∈	∈	NOUN
ejpam-3781	273	4	r	r	NOUN
ejpam-3781	273	5	,	,	PUNCT
ejpam-3781	273	6	then	then	ADV
ejpam-3781	273	7	m	m	VERB
ejpam-3781	273	8	∗m1	∗m1	PUNCT
ejpam-3781	273	9	=	=	SYM
ejpam-3781	273	10	f(m1)m	f(m1)m	PROPN
ejpam-3781	274	1	=	=	PUNCT
ejpam-3781	274	2	λm1	λm1	PROPN
ejpam-3781	274	3	m	m	NOUN
ejpam-3781	274	4	=	=	ADJ
ejpam-3781	274	5	λmm1	λmm1	PROPN
ejpam-3781	274	6	.	.	PUNCT
ejpam-3781	275	1	put	put	VERB
ejpam-3781	275	2	m1	m1	NOUN
ejpam-3781	275	3	=	=	PUNCT
ejpam-3781	275	4	1	1	NUM
ejpam-3781	275	5	λm	λm	X
ejpam-3781	275	6	.	.	PUNCT
ejpam-3781	276	1	then	then	ADV
ejpam-3781	276	2	m	m	VERB
ejpam-3781	276	3	∗m1	∗m1	X
ejpam-3781	276	4	=	=	SYM
ejpam-3781	277	1	1	1	X
ejpam-3781	277	2	.	.	X
ejpam-3781	277	3	define	define	VERB
ejpam-3781	277	4	ψ	ψ	X
ejpam-3781	277	5	:	:	PUNCT
ejpam-3781	277	6	(	(	PUNCT
ejpam-3781	277	7	m,+	m,+	INTJ
ejpam-3781	277	8	,	,	PUNCT
ejpam-3781	277	9	∗)→	∗)→	NUM
ejpam-3781	277	10	(	(	PUNCT
ejpam-3781	277	11	r,+	r,+	NUM
ejpam-3781	277	12	,	,	PUNCT
ejpam-3781	277	13	·	·	PUNCT
ejpam-3781	277	14	)	)	PUNCT
ejpam-3781	277	15	by	by	ADP
ejpam-3781	277	16	ψ(m	ψ(m	NOUN
ejpam-3781	277	17	)	)	PUNCT
ejpam-3781	277	18	=	=	SYM
ejpam-3781	277	19	λm	λm	ADP
ejpam-3781	277	20	for	for	ADP
ejpam-3781	277	21	all	all	DET
ejpam-3781	277	22	m	m	NOUN
ejpam-3781	277	23	∈m	∈m	NOUN
ejpam-3781	277	24	.	.	PUNCT
ejpam-3781	278	1	then	then	ADV
ejpam-3781	278	2	(	(	PUNCT
ejpam-3781	278	3	m,+	m,+	INTJ
ejpam-3781	278	4	,	,	PUNCT
ejpam-3781	278	5	∗	∗	NOUN
ejpam-3781	278	6	)	)	PUNCT
ejpam-3781	278	7	'	'	PUNCT
ejpam-3781	278	8	(	(	PUNCT
ejpam-3781	278	9	r,+	r,+	NUM
ejpam-3781	278	10	,	,	PUNCT
ejpam-3781	278	11	·	·	PUNCT
ejpam-3781	278	12	)	)	PUNCT
ejpam-3781	278	13	.	.	PUNCT
ejpam-3781	279	1	conversely	conversely	ADV
ejpam-3781	279	2	suppose	suppose	VERB
ejpam-3781	279	3	that	that	SCONJ
ejpam-3781	279	4	m	m	VERB
ejpam-3781	279	5	=	=	X
ejpam-3781	279	6	{	{	PUNCT
ejpam-3781	279	7	0	0	NUM
ejpam-3781	279	8	}	}	PUNCT
ejpam-3781	279	9	or	or	CCONJ
ejpam-3781	279	10	(	(	PUNCT
ejpam-3781	279	11	mf	mf	X
ejpam-3781	279	12	,	,	PUNCT
ejpam-3781	279	13	+	+	PROPN
ejpam-3781	279	14	,	,	PUNCT
ejpam-3781	279	15	∗	∗	NOUN
ejpam-3781	279	16	)	)	PUNCT
ejpam-3781	279	17	'	'	PUNCT
ejpam-3781	279	18	(	(	PUNCT
ejpam-3781	279	19	r,+	r,+	NUM
ejpam-3781	279	20	,	,	PUNCT
ejpam-3781	279	21	·	·	PUNCT
ejpam-3781	279	22	)	)	PUNCT
ejpam-3781	279	23	.	.	PUNCT
ejpam-3781	280	1	since	since	SCONJ
ejpam-3781	280	2	the	the	DET
ejpam-3781	280	3	ring	ring	NOUN
ejpam-3781	280	4	{	{	PUNCT
ejpam-3781	280	5	0	0	NUM
ejpam-3781	280	6	}	}	PUNCT
ejpam-3781	280	7	is	be	AUX
ejpam-3781	280	8	commutative	commutative	ADJ
ejpam-3781	280	9	and	and	CCONJ
ejpam-3781	280	10	since	since	SCONJ
ejpam-3781	280	11	any	any	DET
ejpam-3781	280	12	ring	ring	NOUN
ejpam-3781	280	13	isomorphic	isomorphic	ADJ
ejpam-3781	280	14	to	to	ADP
ejpam-3781	280	15	(	(	PUNCT
ejpam-3781	280	16	r,+	r,+	NUM
ejpam-3781	280	17	,	,	PUNCT
ejpam-3781	280	18	·	·	PUNCT
ejpam-3781	280	19	)	)	PUNCT
ejpam-3781	280	20	is	be	AUX
ejpam-3781	280	21	commutative	commutative	ADJ
ejpam-3781	280	22	,	,	PUNCT
ejpam-3781	280	23	the	the	DET
ejpam-3781	280	24	converse	converse	NOUN
ejpam-3781	280	25	is	be	AUX
ejpam-3781	280	26	clear	clear	ADJ
ejpam-3781	280	27	.	.	PUNCT
ejpam-3781	281	1	acknowledgements	acknowledgement	NOUN
ejpam-3781	281	2	the	the	DET
ejpam-3781	281	3	authors	author	NOUN
ejpam-3781	281	4	are	be	AUX
ejpam-3781	281	5	thankful	thankful	ADJ
ejpam-3781	281	6	to	to	ADP
ejpam-3781	281	7	prof.i.ramabhadra	prof.i.ramabhadra	PROPN
ejpam-3781	281	8	sarma	sarma	PROPN
ejpam-3781	281	9	for	for	ADP
ejpam-3781	281	10	his	his	PRON
ejpam-3781	281	11	valuable	valuable	ADJ
ejpam-3781	281	12	comments	comment	NOUN
ejpam-3781	281	13	and	and	CCONJ
ejpam-3781	281	14	suggestions	suggestion	NOUN
ejpam-3781	281	15	.	.	PUNCT
ejpam-3781	282	1	references	reference	NOUN
ejpam-3781	282	2	134	134	NUM
ejpam-3781	282	3	references	reference	NOUN
ejpam-3781	282	4	[	[	X
ejpam-3781	282	5	1	1	NUM
ejpam-3781	282	6	]	]	PUNCT
ejpam-3781	282	7	o.	o.	NOUN
ejpam-3781	282	8	attagun	attagun	NOUN
ejpam-3781	282	9	and	and	CCONJ
ejpam-3781	282	10	h.altindis	h.altindi	NOUN
ejpam-3781	282	11	f.tasdemir	f.tasdemir	NOUN
ejpam-3781	282	12	,	,	PUNCT
ejpam-3781	282	13	a.	a.	NOUN
ejpam-3781	282	14	equiprime	equiprime	PROPN
ejpam-3781	282	15	n	n	CCONJ
ejpam-3781	282	16	-	-	PUNCT
ejpam-3781	282	17	ideals	ideal	NOUN
ejpam-3781	282	18	of	of	ADP
ejpam-3781	282	19	monogenic	monogenic	ADJ
ejpam-3781	282	20	n	n	CCONJ
ejpam-3781	282	21	-	-	PUNCT
ejpam-3781	282	22	groups	group	NOUN
ejpam-3781	282	23	.	.	PUNCT
ejpam-3781	283	1	hacettepe	hacettepe	ADJ
ejpam-3781	283	2	journal	journal	PROPN
ejpam-3781	283	3	of	of	ADP
ejpam-3781	283	4	mathematics	mathematic	NOUN
ejpam-3781	283	5	and	and	CCONJ
ejpam-3781	283	6	statistics	statistic	NOUN
ejpam-3781	283	7	,	,	PUNCT
ejpam-3781	283	8	40:375–382	40:375–382	PROPN
ejpam-3781	283	9	,	,	PUNCT
ejpam-3781	283	10	2011	2011	NUM
ejpam-3781	283	11	.	.	PUNCT
ejpam-3781	284	1	[	[	X
ejpam-3781	284	2	2	2	NUM
ejpam-3781	284	3	]	]	X
ejpam-3781	284	4	n.	n.	NOUN
ejpam-3781	284	5	groenwald	groenwald	NOUN
ejpam-3781	284	6	.	.	PUNCT
ejpam-3781	285	1	on	on	ADP
ejpam-3781	285	2	thel	thel	PROPN
ejpam-3781	285	3	prime	prime	ADJ
ejpam-3781	285	4	radicals	radical	NOUN
ejpam-3781	285	5	of	of	ADP
ejpam-3781	285	6	nea	nea	PROPN
ejpam-3781	285	7	-	-	PUNCT
ejpam-3781	285	8	rings	ring	NOUN
ejpam-3781	285	9	and	and	CCONJ
ejpam-3781	285	10	near	near	ADP
ejpam-3781	285	11	modules	module	NOUN
ejpam-3781	285	12	.	.	PUNCT
ejpam-3781	286	1	nearrings	nearring	NOUN
ejpam-3781	286	2	,	,	PUNCT
ejpam-3781	286	3	near	near	NOUN
ejpam-3781	286	4	-	-	PUNCT
ejpam-3781	286	5	fields	field	NOUN
ejpam-3781	286	6	and	and	CCONJ
ejpam-3781	286	7	related	related	ADJ
ejpam-3781	286	8	and	and	CCONJ
ejpam-3781	286	9	related	related	ADJ
ejpam-3781	286	10	toppics	toppic	NOUN
ejpam-3781	286	11	,	,	PUNCT
ejpam-3781	286	12	119:42–57	119:42–57	NUM
ejpam-3781	286	13	,	,	PUNCT
ejpam-3781	286	14	2017	2017	NUM
ejpam-3781	286	15	.	.	PUNCT
ejpam-3781	287	1	[	[	X
ejpam-3781	287	2	3	3	X
ejpam-3781	287	3	]	]	X
ejpam-3781	287	4	t.v.n	t.v.n	PROPN
ejpam-3781	287	5	.	.	PUNCT
ejpam-3781	287	6	prasanna	prasanna	PROPN
ejpam-3781	287	7	and	and	CCONJ
ejpam-3781	287	8	a.v	a.v	PROPN
ejpam-3781	287	9	.	.	PROPN
ejpam-3781	287	10	ramakrishna	ramakrishna	PROPN
ejpam-3781	287	11	i.r.b	i.r.b	PROPN
ejpam-3781	287	12	.	.	PUNCT
ejpam-3781	288	1	sarma	sarma	PROPN
ejpam-3781	288	2	,	,	PUNCT
ejpam-3781	288	3	j.madhu	j.madhu	PROPN
ejpam-3781	288	4	sudan	sudan	PROPN
ejpam-3781	288	5	rao	rao	PROPN
ejpam-3781	288	6	.	.	PUNCT
ejpam-3781	289	1	near	near	ADP
ejpam-3781	289	2	relatives	relative	NOUN
ejpam-3781	289	3	of	of	ADP
ejpam-3781	289	4	homoneous	homoneous	ADJ
ejpam-3781	289	5	maps	map	NOUN
ejpam-3781	289	6	.	.	PUNCT
ejpam-3781	290	1	southeast	southeast	ADJ
ejpam-3781	290	2	asian	asian	ADJ
ejpam-3781	290	3	bulletin	bulletin	NOUN
ejpam-3781	290	4	of	of	ADP
ejpam-3781	290	5	mathematics	mathematic	NOUN
ejpam-3781	290	6	,	,	PUNCT
ejpam-3781	290	7	38:543–554	38:543–554	NUM
ejpam-3781	290	8	,	,	PUNCT
ejpam-3781	290	9	2014	2014	NUM
ejpam-3781	290	10	.	.	PUNCT
ejpam-3781	291	1	[	[	X
ejpam-3781	291	2	4	4	NUM
ejpam-3781	291	3	]	]	X
ejpam-3781	291	4	k.d	k.d	PROPN
ejpam-3781	291	5	.	.	PROPN
ejpam-3781	291	6	magill	magill	PROPN
ejpam-3781	291	7	,	,	PUNCT
ejpam-3781	291	8	jr	jr	PROPN
ejpam-3781	291	9	.	.	PUNCT
ejpam-3781	291	10	topological	topological	ADJ
ejpam-3781	291	11	nearrings	nearring	NOUN
ejpam-3781	291	12	whose	whose	DET
ejpam-3781	291	13	additive	additive	ADJ
ejpam-3781	291	14	groups	group	NOUN
ejpam-3781	291	15	are	be	AUX
ejpam-3781	291	16	euclidean	euclidean	ADJ
ejpam-3781	291	17	.	.	PUNCT
ejpam-3781	292	1	mathematik	mathematik	PROPN
ejpam-3781	292	2	,	,	PUNCT
ejpam-3781	292	3	119:281–301	119:281–301	NUM
ejpam-3781	292	4	,	,	PUNCT
ejpam-3781	292	5	1995	1995	NUM
ejpam-3781	292	6	.	.	PUNCT
ejpam-3781	293	1	[	[	X
ejpam-3781	293	2	5	5	X
ejpam-3781	293	3	]	]	X
ejpam-3781	293	4	g.	g.	PROPN
ejpam-3781	293	5	pilz	pilz	PROPN
ejpam-3781	293	6	.	.	PUNCT
ejpam-3781	294	1	near	near	ADP
ejpam-3781	294	2	-	-	PUNCT
ejpam-3781	294	3	rings	ring	NOUN
ejpam-3781	294	4	.	.	PUNCT
ejpam-3781	295	1	north	north	NOUN
ejpam-3781	295	2	-	-	PUNCT
ejpam-3781	295	3	holland	holland	PROPN
ejpam-3781	295	4	mathematical	mathematical	PROPN
ejpam-3781	295	5	studies	study	NOUN
ejpam-3781	295	6	,	,	PUNCT
ejpam-3781	295	7	amsterdam	amsterdam	PROPN
ejpam-3781	295	8	,	,	PUNCT
ejpam-3781	295	9	1983	1983	NUM
ejpam-3781	295	10	.	.	PUNCT
ejpam-3781	296	1	[	[	X
ejpam-3781	296	2	6	6	NUM
ejpam-3781	296	3	]	]	X
ejpam-3781	296	4	j.	j.	PROPN
ejpam-3781	296	5	r.clay	r.clay	PROPN
ejpam-3781	296	6	.	.	PUNCT
ejpam-3781	297	1	nearrings	nearring	NOUN
ejpam-3781	297	2	:	:	PUNCT
ejpam-3781	297	3	genesis	genesis	NOUN
ejpam-3781	297	4	and	and	CCONJ
ejpam-3781	297	5	applications	application	NOUN
ejpam-3781	297	6	.	.	PUNCT
ejpam-3781	298	1	oxford	oxford	PROPN
ejpam-3781	298	2	science	science	PROPN
ejpam-3781	298	3	publications	publication	NOUN
ejpam-3781	298	4	,	,	PUNCT
ejpam-3781	298	5	new	new	PROPN
ejpam-3781	298	6	york	york	PROPN
ejpam-3781	298	7	,	,	PUNCT
ejpam-3781	298	8	1992	1992	NUM
ejpam-3781	298	9	.	.	PUNCT