id	sid	eid	entity	type
ejpam-6493	1	1	european journal	ORG
ejpam-6493	1	2	2025	DATE
ejpam-6493	1	3	18	CARDINAL
ejpam-6493	1	4	3	CARDINAL
ejpam-6493	1	5	6493	DATE
ejpam-6493	1	6	1307-5543	DATE
ejpam-6493	1	7	new york	GPE
ejpam-6493	1	8	petapirak1	GPE
ejpam-6493	1	9	jehan khan1	PERSON
ejpam-6493	1	10	1	CARDINAL
ejpam-6493	1	11	90110	DATE
ejpam-6493	1	12	thailand	GPE
ejpam-6493	4	1	2020	DATE
ejpam-6493	4	2	16y60	CARDINAL
ejpam-6493	4	3	16y99	CARDINAL
ejpam-6493	4	4	theorems 1	ORG
ejpam-6493	7	1	semirings	ORG
ejpam-6493	9	1	1–3	CARDINAL
ejpam-6493	10	1	2003	DATE
ejpam-6493	10	2	4	CARDINAL
ejpam-6493	13	1	2011	DATE
ejpam-6493	13	2	5	CARDINAL
ejpam-6493	15	1	2021	CARDINAL
ejpam-6493	16	1	6	CARDINAL
ejpam-6493	17	1	2024	DATE
ejpam-6493	17	2	sanborisoot	ORG
ejpam-6493	20	1	m. petapirak	PERSON
ejpam-6493	20	2	a. j. khan	PERSON
ejpam-6493	20	3	r. chinram	PERSON
ejpam-6493	21	1	1	CARDINAL
ejpam-6493	21	2	2025	CARDINAL
ejpam-6493	22	1	m. petapirak	PERSON
ejpam-6493	22	2	a. j. khan	PERSON
ejpam-6493	22	3	r. chinram	PERSON
ejpam-6493	23	1	j. pure	PERSON
ejpam-6493	23	2	18	CARDINAL
ejpam-6493	23	3	3	CARDINAL
ejpam-6493	23	4	2025	CARDINAL
ejpam-6493	23	5	6493 2	DATE
ejpam-6493	23	6	congruences and/or ideals	ORG
ejpam-6493	24	1	2023-2025	DATE
ejpam-6493	26	1	2	CARDINAL
ejpam-6493	27	1	1	CARDINAL
ejpam-6493	27	2	4	CARDINAL
ejpam-6493	29	1	bcd	ORG
ejpam-6493	29	2	acd	ORG
ejpam-6493	30	1	abc+ abd	PERSON
ejpam-6493	30	2	r.	NORP
ejpam-6493	30	3	2	CARDINAL
ejpam-6493	30	4	4	CARDINAL
ejpam-6493	32	1	∈	ORG
ejpam-6493	33	1	0xy	CARDINAL
ejpam-6493	35	1	0	CARDINAL
ejpam-6493	35	2	zero	CARDINAL
ejpam-6493	36	1	zero	CARDINAL
ejpam-6493	37	1	3	CARDINAL
ejpam-6493	37	2	4	CARDINAL
ejpam-6493	38	1	∈	ORG
ejpam-6493	38	2	4	CARDINAL
ejpam-6493	38	3	4	CARDINAL
ejpam-6493	39	1	1	CARDINAL
ejpam-6493	39	2	∈	ORG
ejpam-6493	39	3	r2 ∈	PERSON
ejpam-6493	39	4	2	CARDINAL
ejpam-6493	39	5	∈	ORG
ejpam-6493	39	6	r2 ∈	PERSON
ejpam-6493	39	7	3	CARDINAL
ejpam-6493	39	8	∈	ORG
ejpam-6493	39	9	r2 ∈	PERSON
ejpam-6493	39	10	4	CARDINAL
ejpam-6493	39	11	r.	NORP
ejpam-6493	39	12	5	DATE
ejpam-6493	41	1	i.	PERSON
ejpam-6493	41	2	6	DATE
ejpam-6493	42	1	two	CARDINAL
ejpam-6493	42	2	φ	ORG
ejpam-6493	43	1	m. petapirak	PERSON
ejpam-6493	43	2	a. j. khan	PERSON
ejpam-6493	43	3	r. chinram	PERSON
ejpam-6493	44	1	j. pure	PERSON
ejpam-6493	44	2	18	CARDINAL
ejpam-6493	44	3	3	CARDINAL
ejpam-6493	44	4	2025	CARDINAL
ejpam-6493	44	5	6493 3	DATE
ejpam-6493	46	1	∈	ORG
ejpam-6493	46	2	r.	NORP
ejpam-6493	46	3	7	DATE
ejpam-6493	47	1	two	CARDINAL
ejpam-6493	48	1	φ	PERSON
ejpam-6493	48	2	φ	ORG
ejpam-6493	49	1	φ	PERSON
ejpam-6493	49	2	t .	DATE
ejpam-6493	50	1	3	CARDINAL
ejpam-6493	51	1	3.1	CARDINAL
ejpam-6493	52	1	firstly	ORDINAL
ejpam-6493	52	2	5	CARDINAL
ejpam-6493	53	1	8	CARDINAL
ejpam-6493	53	2	5	CARDINAL
ejpam-6493	54	1	1	CARDINAL
ejpam-6493	54	2	2	CARDINAL
ejpam-6493	56	1	q+	PERSON
ejpam-6493	56	2	∈	ORG
ejpam-6493	56	3	q3 ∈	PERSON
ejpam-6493	56	4	+ i)(q3	GPE
ejpam-6493	59	1	∈	ORG
ejpam-6493	60	1	⊆	CARDINAL
ejpam-6493	61	1	q′	PERSON
ejpam-6493	61	2	+ i.	PERSON
ejpam-6493	62	1	5	CARDINAL
ejpam-6493	64	1	1	CARDINAL
ejpam-6493	64	2	0	CARDINAL
ejpam-6493	67	1	0,−5,−10,−15	CARDINAL
ejpam-6493	68	1	0	CARDINAL
ejpam-6493	70	1	0 +	DATE
ejpam-6493	71	1	0,−5,−10,−15	CARDINAL
ejpam-6493	74	1	−1,−6,−11,−16	ORG
ejpam-6493	78	1	−3,−8,−13,−18	PERSON
ejpam-6493	80	1	−4,−9,−14,−19	ORG
ejpam-6493	81	1	0,−4,−6,−8	CARDINAL
ejpam-6493	82	1	j	PERSON
ejpam-6493	85	1	m. petapirak	PERSON
ejpam-6493	85	2	a. j. khan	PERSON
ejpam-6493	85	3	r. chinram	PERSON
ejpam-6493	86	1	j. pure	PERSON
ejpam-6493	86	2	18	CARDINAL
ejpam-6493	86	3	3	CARDINAL
ejpam-6493	86	4	2025	CARDINAL
ejpam-6493	86	5	6493 4	DATE
ejpam-6493	86	6	5	CARDINAL
ejpam-6493	87	1	two	CARDINAL
ejpam-6493	87	2	zero	CARDINAL
ejpam-6493	88	1	φ	PERSON
ejpam-6493	88	2	∈	ORG
ejpam-6493	89	1	⊆	CARDINAL
ejpam-6493	89	2	∈	ORG
ejpam-6493	89	3	0	CARDINAL
ejpam-6493	91	1	2	CARDINAL
ejpam-6493	91	2	0	CARDINAL
ejpam-6493	91	3	3	CARDINAL
ejpam-6493	93	1	0	CARDINAL
ejpam-6493	94	1	φ	ORG
ejpam-6493	95	1	0,−3,−6,−9	CARDINAL
ejpam-6493	96	1	∈	ORG
ejpam-6493	96	2	qa ∈	PERSON
ejpam-6493	97	1	qa ̸= 0	ORG
ejpam-6493	102	1	⊆	CARDINAL
ejpam-6493	103	1	0	CARDINAL
ejpam-6493	104	1	1	CARDINAL
ejpam-6493	104	2	2	CARDINAL
ejpam-6493	104	3	−1	DATE
ejpam-6493	105	1	φ	PERSON
ejpam-6493	107	1	0 + ker(φ),−1 + ker(φ),−2 + ker(φ	DATE
ejpam-6493	107	2	0,−3,−6,−9	CARDINAL
ejpam-6493	112	1	1	CARDINAL
ejpam-6493	112	2	5	CARDINAL
ejpam-6493	113	1	two	CARDINAL
ejpam-6493	113	2	zero	CARDINAL
ejpam-6493	114	1	φ	PERSON
ejpam-6493	114	2	t .	DATE
ejpam-6493	115	1	3.2	CARDINAL
ejpam-6493	116	1	modulo congruences	ORG
ejpam-6493	116	2	10	CARDINAL
ejpam-6493	118	1	∈	ORG
ejpam-6493	118	2	1	CARDINAL
ejpam-6493	118	3	2	CARDINAL
ejpam-6493	118	4	a1b1c1	ORG
ejpam-6493	119	1	∈	ORG
ejpam-6493	119	2	∈	ORG
ejpam-6493	120	1	a+ ρ)⊕	PERSON
ejpam-6493	123	1	φ	ORG
ejpam-6493	125	1	k(φ	PERSON
ejpam-6493	127	1	k(φ	PERSON
ejpam-6493	127	2	r. m. petapirak	PERSON
ejpam-6493	127	3	a. j. khan	PERSON
ejpam-6493	127	4	r. chinram	PERSON
ejpam-6493	128	1	j. pure	PERSON
ejpam-6493	128	2	18	CARDINAL
ejpam-6493	128	3	3	CARDINAL
ejpam-6493	128	4	2025	CARDINAL
ejpam-6493	128	5	6493 5	DATE
ejpam-6493	128	6	9	CARDINAL
ejpam-6493	131	1	3	CARDINAL
ejpam-6493	132	1	4	CARDINAL
ejpam-6493	132	2	φ(a	ORG
ejpam-6493	133	1	∈	ORG
ejpam-6493	133	2	r.	NORP
ejpam-6493	133	3	φ	ORG
ejpam-6493	134	1	k(φ	PERSON
ejpam-6493	134	2	r.	NORP
ejpam-6493	135	1	−1 +k(φ),−2	DATE
ejpam-6493	135	2	+k(φ	PERSON
ejpam-6493	138	1	+k(φ	PERSON
ejpam-6493	143	1	+k(φ	PERSON
ejpam-6493	150	1	3.3	CARDINAL
ejpam-6493	154	1	∈	ORG
ejpam-6493	156	1	4	CARDINAL
ejpam-6493	157	1	0	CARDINAL
ejpam-6493	159	1	1	CARDINAL
ejpam-6493	160	1	0,−5,−10,−15	CARDINAL
ejpam-6493	161	1	0	CARDINAL
ejpam-6493	162	1	0 +	DATE
ejpam-6493	163	1	0,−5,−10,−15	CARDINAL
ejpam-6493	166	1	−1,−6,−11,−16	ORG
ejpam-6493	170	1	−3,−8,−13,−18	PERSON
ejpam-6493	172	1	−4,−9,−14,−19	ORG
ejpam-6493	174	1	example 1	DATE
ejpam-6493	174	2	r. m. petapirak	PERSON
ejpam-6493	174	3	a. j. khan	PERSON
ejpam-6493	174	4	r. chinram	PERSON
ejpam-6493	175	1	j. pure	PERSON
ejpam-6493	175	2	18	CARDINAL
ejpam-6493	175	3	3	CARDINAL
ejpam-6493	175	4	2025	CARDINAL
ejpam-6493	175	5	6493 6	DATE
ejpam-6493	175	6	9	CARDINAL
ejpam-6493	175	7	2	CARDINAL
ejpam-6493	175	8	0,−4,−6,−8	CARDINAL
ejpam-6493	176	1	0	CARDINAL
ejpam-6493	182	1	0,−2,−4,−6	DATE
ejpam-6493	183	1	1	CARDINAL
ejpam-6493	185	1	4	CARDINAL
ejpam-6493	185	2	congruences	ORG
ejpam-6493	185	3	3.2	CARDINAL
ejpam-6493	185	4	3.3	CARDINAL
ejpam-6493	186	1	firstly	ORDINAL
ejpam-6493	186	2	4	CARDINAL
ejpam-6493	186	3	2	CARDINAL
ejpam-6493	188	1	⊆	CARDINAL
ejpam-6493	191	1	2	CARDINAL
ejpam-6493	192	1	1	CARDINAL
ejpam-6493	192	2	zero	CARDINAL
ejpam-6493	193	1	2	CARDINAL
ejpam-6493	193	2	zero	CARDINAL
ejpam-6493	193	3	∈	ORG
ejpam-6493	195	1	3	CARDINAL
ejpam-6493	195	2	∈	ORG
ejpam-6493	196	1	4	CARDINAL
ejpam-6493	196	2	zero	CARDINAL
ejpam-6493	197	1	∈	ORG
ejpam-6493	198	1	1	CARDINAL
ejpam-6493	199	1	r.	NORP
ejpam-6493	199	2	⊆	CARDINAL
ejpam-6493	201	1	∈	ORG
ejpam-6493	203	1	∈	ORG
ejpam-6493	204	1	∈	ORG
ejpam-6493	205	1	2	CARDINAL
ejpam-6493	205	2	∈	ORG
ejpam-6493	206	1	∈	ORG
ejpam-6493	206	2	0	CARDINAL
ejpam-6493	206	3	∈	ORG
ejpam-6493	208	1	a+ i.	PERSON
ejpam-6493	208	2	3	CARDINAL
ejpam-6493	208	3	∈	ORG
ejpam-6493	208	4	∈	ORG
ejpam-6493	209	1	+ i.	PERSON
ejpam-6493	209	2	∈	ORG
ejpam-6493	209	3	a+ u ∈	PERSON
ejpam-6493	209	4	∈	ORG
ejpam-6493	210	1	∈	ORG
ejpam-6493	211	1	∈	ORG
ejpam-6493	212	1	i.	PERSON
ejpam-6493	212	2	4	CARDINAL
ejpam-6493	212	3	3	CARDINAL
ejpam-6493	212	4	1	CARDINAL
ejpam-6493	212	5	2	CARDINAL
ejpam-6493	213	1	r.	NORP
ejpam-6493	214	1	∈	ORG
ejpam-6493	218	1	11	CARDINAL
ejpam-6493	219	1	2	CARDINAL
ejpam-6493	221	1	m. petapirak	PERSON
ejpam-6493	221	2	a. j. khan	PERSON
ejpam-6493	221	3	r. chinram	PERSON
ejpam-6493	222	1	j. pure	PERSON
ejpam-6493	222	2	18	CARDINAL
ejpam-6493	222	3	3	CARDINAL
ejpam-6493	222	4	2025	CARDINAL
ejpam-6493	222	5	6493 7	DATE
ejpam-6493	222	6	9	CARDINAL
ejpam-6493	223	1	φ	PERSON
ejpam-6493	223	2	a+ ck(i	PERSON
ejpam-6493	223	3	∈	ORG
ejpam-6493	223	4	first	ORDINAL
ejpam-6493	223	5	∈	ORG
ejpam-6493	225	1	⊆	CARDINAL
ejpam-6493	225	2	a+ck(i	ORG
ejpam-6493	226	1	φ	ORG
ejpam-6493	227	1	φ	PERSON
ejpam-6493	228	1	∈	ORG
ejpam-6493	231	1	a+ck(i	ORG
ejpam-6493	236	1	∈ i.	ORG
ejpam-6493	240	1	+ i.	PERSON
ejpam-6493	240	2	φ	ORG
ejpam-6493	240	3	one	CARDINAL
ejpam-6493	241	1	φ	ORG
ejpam-6493	242	1	k(φ	PERSON
ejpam-6493	244	1	r.	NORP
ejpam-6493	245	1	φ	GPE
ejpam-6493	246	1	∈	ORG
ejpam-6493	248	1	φ	ORG
ejpam-6493	253	1	∈	ORG
ejpam-6493	253	2	r.	NORP
ejpam-6493	253	3	+k(φ	GPE
ejpam-6493	264	1	φ(a	ORG
ejpam-6493	267	1	0r	CARDINAL
ejpam-6493	267	2	0	CARDINAL
ejpam-6493	268	1	φ	PERSON
ejpam-6493	269	1	0	CARDINAL
ejpam-6493	271	1	∈	ORG
ejpam-6493	271	2	0	CARDINAL
ejpam-6493	271	3	r.	NORP
ejpam-6493	273	1	0r	CARDINAL
ejpam-6493	273	2	∈	ORG
ejpam-6493	273	3	∈	ORG
ejpam-6493	274	1	φ(a	ORG
ejpam-6493	274	2	0	CARDINAL
ejpam-6493	274	3	0	CARDINAL
ejpam-6493	275	1	φ	ORG
ejpam-6493	278	1	0	CARDINAL
ejpam-6493	279	1	∈	ORG
ejpam-6493	280	1	∈	ORG
ejpam-6493	280	2	r.	NORP
ejpam-6493	280	3	0	CARDINAL
ejpam-6493	281	1	rab ∈	PERSON
ejpam-6493	282	1	abr ∈ ker(φ	ORG
ejpam-6493	283	1	r.	NORP
ejpam-6493	283	2	∈	ORG
ejpam-6493	283	3	∈	ORG
ejpam-6493	283	4	∈	ORG
ejpam-6493	284	1	0	CARDINAL
ejpam-6493	285	1	0	CARDINAL
ejpam-6493	286	1	∈	ORG
ejpam-6493	287	1	r.	NORP
ejpam-6493	287	2	0r	CARDINAL
ejpam-6493	287	3	0	CARDINAL
ejpam-6493	287	4	φ	PERSON
ejpam-6493	288	1	0	CARDINAL
ejpam-6493	289	1	φ	PERSON
ejpam-6493	290	1	∈	ORG
ejpam-6493	291	1	first	ORDINAL
ejpam-6493	291	2	3	CARDINAL
ejpam-6493	292	1	φ	ORG
ejpam-6493	294	1	φ	PERSON
ejpam-6493	296	1	φ	ORG
ejpam-6493	298	1	φ(s	GPE
ejpam-6493	298	2	∈	ORG
ejpam-6493	298	3	r.	NORP
ejpam-6493	298	4	three	CARDINAL
ejpam-6493	298	5	r.	NORP
ejpam-6493	299	1	∈	ORG
ejpam-6493	300	1	φ(a	ORG
ejpam-6493	304	1	m. petapirak	PERSON
ejpam-6493	304	2	a. j. khan	PERSON
ejpam-6493	304	3	r. chinram	PERSON
ejpam-6493	305	1	j. pure	PERSON
ejpam-6493	305	2	18	CARDINAL
ejpam-6493	305	3	3	CARDINAL
ejpam-6493	305	4	2025	CARDINAL
ejpam-6493	305	5	6493 8 of 9	DATE
ejpam-6493	305	6	φ	ORG
ejpam-6493	308	1	+ s2	PERSON
ejpam-6493	309	1	+ ker(φ))(s2 + ker(φ))(s3 + ker(φ	PERSON
ejpam-6493	319	1	∈	ORG
ejpam-6493	322	1	5	CARDINAL
ejpam-6493	323	1	4	CARDINAL
ejpam-6493	323	2	φ(a	ORG
ejpam-6493	324	1	∈	ORG
ejpam-6493	324	2	r.	NORP
ejpam-6493	324	3	φ	ORG
ejpam-6493	326	1	r.	NORP
ejpam-6493	327	1	−1 +	DATE
ejpam-6493	330	1	−3 + ker(φ	PERSON
ejpam-6493	335	1	1	CARDINAL
ejpam-6493	336	1	0r	CARDINAL
ejpam-6493	336	2	0	CARDINAL
ejpam-6493	336	3	φ	PERSON
ejpam-6493	337	1	0	CARDINAL
ejpam-6493	338	1	φ	PERSON
ejpam-6493	339	1	5	CARDINAL
ejpam-6493	339	2	modulo congruences	ORG
ejpam-6493	340	1	m. petapirak	PERSON
ejpam-6493	340	2	a. j. khan	PERSON
ejpam-6493	340	3	r. chinram	PERSON
ejpam-6493	341	1	j. pure	PERSON
ejpam-6493	341	2	18	CARDINAL
ejpam-6493	341	3	3	CARDINAL
ejpam-6493	341	4	2025	CARDINAL
ejpam-6493	341	5	6493 9	DATE
ejpam-6493	345	1	no.1	ORG
ejpam-6493	346	1	1	CARDINAL
ejpam-6493	346	2	j. allen	PERSON
ejpam-6493	348	1	american	NORP
ejpam-6493	348	2	24:463–465	CARDINAL
ejpam-6493	348	3	1969	DATE
ejpam-6493	349	1	2	CARDINAL
ejpam-6493	349	2	s. e. atani	PERSON
ejpam-6493	349	3	a. g. garfami	PERSON
ejpam-6493	351	1	chiang mai journal of science	ORG
ejpam-6493	351	2	40(1):77–82	CARDINAL
ejpam-6493	351	3	2013	DATE
ejpam-6493	352	1	3	CARDINAL
ejpam-6493	352	2	d. r.	NORP
ejpam-6493	354	1	american	NORP
ejpam-6493	354	2	24:463–465	CARDINAL
ejpam-6493	354	3	1970	DATE
ejpam-6493	355	1	4	CARDINAL
ejpam-6493	355	2	t. k. dutta	PERSON
ejpam-6493	355	3	s. kar	PERSON
ejpam-6493	357	1	icm	ORG
ejpam-6493	357	2	343–355	PRODUCT
ejpam-6493	357	3	2003	DATE
ejpam-6493	358	1	5	CARDINAL
ejpam-6493	358	2	j. n.	PERSON
ejpam-6493	358	3	k. j. ingale	PERSON
ejpam-6493	360	1	kyungpook mathematical journal	ORG
ejpam-6493	360	2	51(1):69–76, 2011	DATE
ejpam-6493	361	1	6	CARDINAL
ejpam-6493	361	2	nagi reddy	PERSON
ejpam-6493	361	3	g. shobhalatha	PERSON
ejpam-6493	363	1	indian journal of science and technology	ORG
ejpam-6493	363	2	2021	DATE
ejpam-6493	364	1	7	CARDINAL
ejpam-6493	364	2	j. sanborisoot	PERSON
ejpam-6493	364	3	p. p. n. ayutthaya	PERSON
ejpam-6493	365	1	ternary ring congruences	ORG
ejpam-6493	366	1	discussiones mathematicae	ORG
ejpam-6493	366	2	44:43–55	DATE
ejpam-6493	366	3	2024	DATE
ejpam-6493	367	1	8	CARDINAL
ejpam-6493	367	2	n. sarasit	PERSON
ejpam-6493	367	3	r. chinram	PERSON
ejpam-6493	367	4	a. rattana	PERSON
ejpam-6493	369	1	36(4):497–508	CARDINAL
ejpam-6493	369	2	2023	CARDINAL
ejpam-6493	370	1	9	CARDINAL
ejpam-6493	370	2	t. changphas	ORG
ejpam-6493	371	1	one	CARDINAL
ejpam-6493	372	1	international journal of mathematics and computer science	ORG
ejpam-6493	372	2	19(2):403–409	DATE
ejpam-6493	372	3	2024	DATE
ejpam-6493	373	1	10	CARDINAL
ejpam-6493	373	2	r. chinram	PERSON
ejpam-6493	375	1	26(2):120–131	ORG
ejpam-6493	375	2	2025	DATE
ejpam-6493	376	1	11	CARDINAL
ejpam-6493	376	2	a. j. khan	PERSON
ejpam-6493	376	3	m. petapirak	PERSON
ejpam-6493	376	4	r. chinram	PERSON
ejpam-6493	378	1	european journal	ORG
ejpam-6493	378	2	18(2):6096	CARDINAL
ejpam-6493	378	3	2025	DATE