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Mathematical Problems of Computer Science 39, 94{100, 2013.

Radicals and P r er adicals in the Categor y of M odules

over All Rings

Gr ig o r G. E m in -Te r ya n ( E m in )

Institute for Informatics and Automation Problems of NAS RA

e-mail: grigoremin@rambler.ru

Abstract

Let Mod be a category whose objects are all possible pairs (A; U), where U is an
associative ring, A is a right U-module (not unitary in the general case) and the set of
morphisms of module (A; U) to module (B; V ) consists of pairs of mappings ('A; 'U ),
where 'A or 'U , respectively, is a homomorphism of Abelian group A to Abelian group
B (ring U to ring V ), where (a ¢ u)'A = a 'A ¢ u 'U , a 2 A, u 2 U. This pair of
mappings is called a homomorphism of module (A; U) to module (B; V ). It is proved
that strict radicals of Mod in the sense of Kurosh are described by means of systems of
strict radicals of the category of associative rings As and categories of right U-modules
Mod ¡ U, U 2 As. It turned out that a wider classes of preradicals and radicals of
Mod can also be described by means of systems of preradicals and radicals of As and
Mod ¡ U, U 2 As, respectively, which satisfy some conditions.

Keywords: Module, associative ring, Abelian group, category, radical, preradical,
torsion, ideally hereditary radical, strongly hereditary radical.

L e t u s c o n s id e r t h e c a t e g o r y M od, wh o s e o b je c t s a r e a ll p o s s ib le p a ir s ( A; U ) , wh e r e A
is a r ig h t U-m o d u le o ve r a n a s s o c ia t ive r in g U, a n d t h e m o r p h is m s a r e p a ir s ( 'A; 'U ) o f
h o m o m o r p h is m s s a t is fyin g t h e c o n d it io n ( a ¢ u ) 'A = a 'A ¢ u 'U , a 2 A, u 2 U [1 ].

De¯nition 1: W e will say that a preradical r is de¯ned in the Mod, if an ideal r ( A; U )
matches to each module ( A; U ) 2 Mod such that r ( A; U ) ( 'A; 'U ) µ r ( B; V ) for each
homomorphism ( 'A; 'U ) : ( A; U ) ! ( B; V ) of M od [2], [3].

In o t h e r wo r d s , t h e p r e r a d ic a l r is a n o r m a l s u b fu n c t o r o f t h e id e n t it y fu n c t o r IdM od :
Mod ! M od. In e a c h m o d u le ( A; U ) 2 Mod it s in g le s o u t a n id e a l r ( A; U ) s u c h t h a t fo r
e ve r y h o m o m o r p h is m ( 'A; 'U ) : ( A; U ) ! ( B; V ) t h e d ia g r a m :

( A; U ) ¡! ( B; V )
" "

r ( A; U ) ¡! r ( B; V )

is c o m m u t a t ive .
A s s u m e t h a t in t h e c a t e g o r y Mod s o m e p r e r a d ic a l r is s p e c ī e d . Co n s id e r a fu ll s u b c a t -

e g o r y Mod( As) o f M od wh o s e o b je c t s a r e a ll m o d u le s o f t h e fo r m ( 0 ; U ) . A n y id e a ls a n d

9 4



G. Emin-Teryan (Emin) 9 5

a n y h o m o m o r p h ic im a g e s o f o b je c t s fr o m Mod ( As) t h e m s e lve s lie in s u b c a t e g o r y Mod( As) ,
a n d h e n c e , t h e p r e r a d ic a l r in d u c e s a c o m p le t e ly d e t e r m in a t e p r e r a d ic a l R in s u b c a t e g o r y
Mod( As) . Th is m e a n s t h a t a p r e r a d ic a l r d e ¯ n e s a c o m p le t e ly d e t e r m in a t e p r e r a d ic a l R in
c a t e g o r y As o f a s s o c ia t ive r in g a n d r ( 0 ; U ) = ( 0 ; R( U ) ) .

Lemma 1: L et r be any preradical in Mod, (A,U) be any module and r ( A; U ) = ( A0; U 0 ) .
Then U 0 = R( U ) , where R is a preradical in As induced of preradical r of Mod.

P r oof. S in c e r is a p r e r a d ic a l a n d ( 0 ; U ) is a s u b m o d u le o f ( A; U ) t h e n ( 0 ; R ( U ) ) = r ( 0 ; U ) µ
r ( A; U ) = ( A0; U 0 ) . H e n c e R( U ) µ U 0. B u t fo r h o m o m o r p h is m ( 0 ; 1 U ) : ( A; U ) ! ( 0 ; U ) we
h a ve ( A0; U 0 ) ( 0 ; 1 U ) = r ( A; U ) ( 0 ; 1 U ) µ r ( 0 ; U ) = ( 0 ; R( U ) ) , i.e ., U 0 µ R ( U ) . L e m m a 1 : is
p r o ve d .

R e c a ll t h a t t h e s u b m o d u le ( A0; U 0 ) o f ( A; U ) will b e a n id e a l o f ( A; U ) if a n d o n ly if U 0

is a n id e a l o f t h e r in g U a n d t h e in c lu s io n s A ¢ U 0 µ A0 a n d A0 ¢ U µ A0 a r e o b s e r ve d [1 ].
A s s u m e r is a n y p r e r a d ic a l o f Mod, ( A; U ) is a n a n y m o d u le a n d r ( A; U ) = ( A0; U 0 ) .

S in c e ( A0; U 0 ) is a n id e a l o f ( A; U ) , t h e n A0 is a U -s u b m o d u le o f U-m o d u le A.
N o w s h o w t h a t m a t c h in g s u b m o d u le A0 t o e a c h U -m o d u le A d e ¯ n e s t h e p r e r a d ic a l Ru in

t h e c a t e g o r y o f r ig h t U-m o d u le s Mod ¡ U.
A s s u m e A a n d B a r e U-m o d u le s , ' : A ! B is a n U-h o m o m o r p h is m a n d r ( A; U ) =

( A0; U 0 ) , r ( B; U ) = ( B0; U 0 ) , wh e r e U 0 = R( U ) b y L e m m a 1 :. W e 'll s h o w t h a t A0' µ B0.
In d e e d , s in c e ( '; 1 U ) : ( A; U ) ! ( B; U ) is a h o m o m o r p h is m in Mod, t h e n ( A0; U 0 ) ( '; 1 U ) =
r ( A; U ) ( '; 1 U ) µ r ( B; U ) = ( B0; U 0 ) . H e n c e A0' µ B0. Th u s , fo r a n y U -m o d u le A, if
r ( A; U ) = ( A0; U 0 ) , m a t c h in g o f U-s u b m o d u le Ru ( A ) = A

0 t o e a c h U-m o d u le A d e ¯ n e s t h e
p r e r a d ic a l Ru in t h e c a t e g o r y Mod ¡ U. Th u s , we h a ve t h e fo llo win g le m m a .

Lemma 2: E ach preradical r of the category M od induces a completely determinate pre-
radical R in As and preradicals RU in the categories M od ¡ U ( U 2 As) such that
r ( A; U ) = ( RU ( A ) ; R( U ) ) for any module ( A; U ) of the category M od.

L e t u s c o n s id e r t h e a c t io n o f t h e r in g h o m o m o r p h is m ' : U ! V . A n y r ig h t V -m o d u le B
b e c o m e s a r ig h t U -m o d u le if t h e a c t io n o f t h e o p e r a t o r s is d e ¯ n e d a s fo llo ws : a ± u = a ( u') ,
we will s a y t h a t t h e m o d u le B is c o n ve r t e d in t o t h e U-m o d u le B' b y wit h d r a wa l a lo n g '.

A s s u m e R is a n y p r e r a d ic a l o f As a n d RU ( U 2 As) a r e p r e r a d ic a ls o f t h e c a t e g o r ie s
Mod ¡ U. Co n s id e r t h e s ys t e m o f p r e r a d ic a ls fR; RU j U 2 Asg.

De¯nition 2: The system of preradicals fR; RU j U 2 Asg will be called a matched system
of preradicals, if it satis¯es the following conditions:

(P 1) A ¢ R( U ) µ RU ( A ) for each U-module A

(P 2) RU ( B' ) µ RV ( B ) ) for each V -module B and each homomorphism ' : U ! V of rings,
where B' is the conversion of V -module B into an U-module by the withdrawal along
'.

T heor em 1: Assume that the preradical r is speci¯ed in M od. Then in As it induces
the preradical R and in each category Mod ¡ U ( U 2 As) it induces preradicals RU such
that the system of preradicals fR; RU j U 2 Asg is matched. Conversely, every matched
system of preradicals fR; RU j U 2 Asg speci¯es a completely determinate preradical r in
Mod such that r ( A; U ) = ( RU ( A) ; R ( U ) ) for each ( A; U ) 2 Mod. There exists a one-to-one
correspondence between all the preradicals of M od and all the matched systems of preradicals.



9 6 Radicals and Preradicals in the Category of Modules over All Rings

P r oof. A s s u m e t h a t r is a p r e r a d ic a l o f Mod. Th e n , b y L e m m a 2 :, we h a ve a c o m -
p le t e ly d e t e r m in a t e s ys t e m o f p r e r a d ic a ls fR; RU j U 2 Asg. S in c e fo r e a c h ( A; U ) 2 Mod,
r ( A; U ) = ( RU ( A ) ; R ( U ) ) is a n id e a l o f m o d u le ( A; U ) , h e n c e A ¢ R( U ) µ RU ( A ) . Th u s , t h e
c o n d it io n ( P 1 ) is s a t is ¯ e d .

A s s u m e t h a t B is a V -m o d u le , ' : U ! V is a h o m o m o r p h is m o f r in g s , a n d B' is
t h e c o n ve r s io n o f V -m o d u le B in t o a n U -m o d u le b y wit h d r a wa l a lo n g '. A s t h e p a ir
( 1 ; ') : ( B'; U ) ! ( B; V ) is a h o m o m o r p h is m in Mod, t h e n ( RU ( B' ) ; R ( U ) ) ( 1 ; ') =
r ( B'; U ) ( 1 ; ') µ r ( B; V ) = ( RV ( B ) ; R ( V ) ) .

Co n ve r s e ly, a s s u m e t h a t we a r e g ive n a m a t c h e d s ys t e m o f p r e r a d ic a ls fR; RU j U 2 Asg
a n d ( A; U ) 2 Mod. S in c e R ( U ) is a n id e a l o f r in g U, RU ( A) is a s u b m o d u le o f U-m o d u le A
a n d A ¢ R( U ) µ RU ( A) b y c o n d it io n ( P 1 ) , t h e n t h e s u b m o d u le ( RU ( A) ; R( U ) ) is a n id e a l o f
t h e m o d u le ( A; U ) .

L e t ( '; Ã ) : ( A; U ) ! ( B; V ) b e a n y h o m o m o r p h is m . It is c le a r t h a t R ( U ) Ã µ R ( V ) .
On t h e o t h e r h a n d , it is e a s ily s e e n t h a t t h e h o m o m o r p h is m ( '; Ã ) = ( '; 1 ) ¢ ( 1 ; Ã ) , wh e r e
( '; 1 ) : ( A; U ) ! ( BÃ; U ) a n d ( 1 ; Ã ) : ( BÃ; U ) ! ( B; V ) . S in c e RU is a p r e r a d ic a l in Mod¡U
a n d ( '; 1 ) is a n U -h o m o m o r p h is m , t h e n RU ( A) ' µ RU ( BÃ ) . A n d fr o m t h e c o n d it io n ( P 2 )
it fo llo ws t h a t RU ( BÃ ) µ RV ( B ) . H e n c e RU ( A) ' µ RV ( B ) .

Th e t h ir d a s s e r t io n o f t h is t h e o r e m fo llo ws fr o m t h e ¯ r s t t wo o n e s , s in c e t h e e xp r e s -
s io n s g ive n in t h e t h e o r e m a b o ve a r e m u t u a lly in ve r s ive . Th u s , t h e p r o o f o f t h e t h e o r e m is
c o m p le t e d .
Remar k 1 It is e a s y t o ve r ify [2 ] t h a t if r is a p r e r a d ic a l in M od, t h e n r ( A; U ) in c lu d e s a ll
t h e r-s u b m o d u le s o f t h e m o d u le ( A; U ) fo r e a c h m o d u le ( A; U ) . A ls o t h e c la s s o f r-r a d ic a l
m o d u le s is e n c lo s e d u n d e r t h e o p e r a t io n o f h o m o m o r p h ic im a g e s .

De¯nition 3: A preradical r of Mod is called a radical if r ( ( A; U ) =r ( A; U ) ) = ( 0 ; 0 ) for
each module ( A; U ) [4].

S im ila r t o L e m m a s 1 : a n d 2 :, o n e m a y p r o ve t h e fo llo win g le m m a .

Lemma 3: L et r be a radical in Mod, ( A; U ) be some module and let r ( A; U ) = ( A0; U 0 ) .
Then U 0 = R ( U ) , where R is the radical in As induced of the radical r of M od.

Lemma 4: E very radical r of the category Mod induces a completely determinate radical R
in As and radicals Ru in categories Mod ¡ U ( U 2 As) such that r ( A; U ) = ( RU ( A) ; R( U ) )
for any module ( A; U ) of the category Mod.

A s s u m e t h a t R is a r a d ic a l o f As a n d RU ( U 2 As) a r e r a d ic a ls o f c a t e g o r ie s M od ¡ U.
Co n s id e r t h e s ys t e m o f r a d ic a ls fR; RU j U 2 Asg.

De¯nition 4: The system of radicals fR; RU j U 2 Asg is called a matched system of
radicals if it satis¯es the following conditions:

(P 1) A ¢ R ( U ) µ RU ( A ) for each U-module A

(P 2) RU ( B' ) µ RV ( B ) ) for each V -module B and each homomorphism ' : U ! V of rings,
where B' is the conversion of V -module B into an U-module by the withdrawal along
'.



G. Emin-Teryan (Emin) 9 7

(P 3) RU=R(U) ( A) = RU ( A¼ ) ) for each U=R( U ) -module A, where ¼ : U ! U=R ( U ) is a
natural epimorphism and A¼ is the conversion of U=R ( U ) module A into U-module by
withdrawal along ¼.

T heor em 2: Assume that some radical r is speci¯ed in Mod. Then it induces the radical
R in As and radicals RU in each category M od ¡ U, U 2 As such that the system of radicals
fR; RU j U 2 Asg is matched. Conversely, every matched system of radicals fR; RU j U 2
Asg speci¯es a completely determinate radical r in M od such that r ( A; U ) = ( RU ( A) ; R ( U ) )
for each ( A; U ) 2 Mod. There exists a one-to-one correspondence between all the radicals of
Mod and all the matched systems of radicals.

P r oof. A s s u m e t h a t r is a r a d ic a l in Mod a n d fR; RU j U 2 Asg is a c o m p le t e ly d e t e r m in e d
s ys t e m o f p r e r a d ic a ls . Ob vio u s ly, if r is a r a d ic a l in Mod, R is a r a d ic a l in As. Mo r e o ve r ,
( RU=R(U) ( A=RU ( A ) ) ; R( U=R ( U ) ) ) = r ( ( A; U ) =r ( A; U ) ) = ( 0 ; 0 ) , i.e . RU=R(U ) ( A=RU ( A ) ) = 0
fo r e a c h ( A; U ) 2 Mod. H e n c e , a s it fo llo ws fr o m c o n d it io n ( P 2 ) RU ( A=RU ( A ) ) = 0 fo r
e ve r y U-m o d u le A.

L e t ¼ : U ! U=R( U ) b e a n a t u r a l e p im o r p h is m a n d A a n a r b it r a r y U=R ( U ) -m o d u le .
Co n s id e r U -m o d u le A¼ in wh ic h a ± u = a ¤ ( u + R ( U ) ) , a 2 A, u 2 U b y d e ¯ n it io n .
Ob vio u s ly, e ve r y s u b m o d u le o f U-m o d u le A¼ is a s u b m o d u le o f U=R ( U ) -m o d u le A. S in c e
A ± R ( U ) = 0 , t h e r e ve r s e c o n ve r s io n , i.e . c o n ve r s io n t o o p e r a t io n a ¤ ( u + R( U ) ) = a ± u,
a 2 A, u 2 U a ls o h o ld s . Th e r e fo r e U-m o d u le A¼ is t r a n s fo r m e d in t o a U=R( U ) -
m o d u le A, a n d t h e s u b m o d u le s a n d fa c t o r m o d u le s o f U-m o d u le A¼ a r e t r a n s fo r m e d in t o
s u b m o d u le s a n d fa c t o r m o d u le s o f U=R ( U ) -m o d u le A. Co n s id e r a n a t u r a l U -e p im o r p h is m
µ : A¼ ! A¼=RU ( A¼ ) . Ob vio u s ly, fo r t r a n s fe r t o ¤ o p e r a t io n t h e m a p p in g µ b e c o m e s
a n U=R ( U ) -h o m o m o r p h is m . Th u s , b e c a u s e RU=R(U) is a p r e r a d ic a l o f Mod ¡ U=RU we
h a ve RU=R(U ) ( A) µ µ RU=R(U) ( A¼=RU ( A¼ ) ) . On t h e o t h e r h a n d , a s r is a r a d ic a l o f M od,
t h e n in vie w o f a s s e r t io n a b o ve RU=R(U ) ( A¼=RU ( A¼ ) ) = 0 . H e n c e RU=R(U ) ( A) µ = 0 ,
i.e . RU=R(U) ( A) µ RU ( A¼ ) fo r e a c h U=R ( U ) -m o d u le A. Th e r e fo r e , b y c o n d it io n ( P 2 )
RU ( A¼ ) µ RU=R(U) ( A ) fo r e a c h U=R ( U ) -m o d u le A.

Co n ve r s e ly, le t ( A; U ) b e a r a n d o m m o d u le o f Mod. S in c e R is a r a d ic a l o f As, t h e n
r ( ( A; U ) =r ( A; U ) ) = ( RU=R(U) ( A=RU ( A ) ; 0 ) ) . A n a p p lic a t io n o f c o n d it io n ( P 3 ) t o U=R( U ) -
m o d u le A=RU ( A ) yie ld s RU=R(U) ( A=RU ( A ) ) = RU ( A=RU ( A) ) = 0 , a s RU is a r a d ic a l o f
Mod ¡ U . Th e t h ir d a s s e r t io n in t h is t h e o r e m fo llo ws fr o m Th e o r e m 1 :. Th e t h e o r e m is
p r o ve d .

De¯nition 5: A preradical r of M od is called idempotent, if r ( r ( A; U ) ) = r ( A; U ) for each
module ( A; U ) .

T heor em 3: The matched system of preradicals fR; RU j U 2 Asg de¯nes an idempotent
preradical of the category Mod if and only if all preradicals of that system are idempotent
and the condition

(P 4) RU ( A) = RR(U) ( A)

holds for each U-module A.

P r oof. A s s u m e t h a t r is a n id e m p o t e n t p r e r a d ic a l o f M od a n d fR; RU j U 2 Asg is t h e
c o r r e s p o n d in g m a t c h e d s ys t e m o f p r e r a d ic a ls . B e c a u s e r ( 0 ; U ) = ( 0 ; R ( U ) ) fo r a n y ( 0 ; U ) 2
Mod( As) a n d r is id e m p o t e n t , R will b e a ls o id e m p o t e n t . Th e r e fo r e , t h e id e m p o t e n c e o f r



9 8 Radicals and Preradicals in the Category of Modules over All Rings

m e a n s t h a t ( RU ( A) ; R ( U ) ) = ( RR(U) ( RU ( A ) ) ; R( U ) ) , i.e . RU ( A) = RR(U) ( RU ( A ) fo r e a c h
m o d u le ( A; U ) o f M od.

On t h e o t h e r h a n d , it fo llo ws fr o m ( P 2 ) t h a t RR(U ) ( RU ( A ) ) µ RU ( RU ( A ) ) . H e n c e
RU ( A ) = RR(U) ( RU ( A) ) µ RU ( RU ( A ) ) µ RU ( A ) , i.e . RU ( RU ( A ) ) = RU ( A) fo r e a c h U-
m o d u le A.

A s we h a ve a lr e a d y s e e n if r is id e m p o t e n t , RU ( A) = RR(U) ( RU ( A) ) fo r e a c h U -m o d u le
A. On t h e o t h e r h a n d fo r e a c h U -m o d u le A, RU ( A) is a R ( U ) -s u b m o d u le o f R( U ) -m o d u le
A, b e c a u s e RU ( A ) ¢ R ( U ) µ A ¢ R( U ) µ RU ( A) b y c o n d it io n ( P 1 ) . H e n c e RR(U ) ( RU ( A ) ) µ
RR(U) ( A) . W e o b t a in t h a t RU ( A ) µ RR(U ) ( A ) . B u t in vie w o f ( P 2 ) RR(U ) ( A) µ RU ( A) .
H e n c e RU ( A ) = RR(U ) ( A ) fo r e a c h U -m o d u le A.

Co n ve r s e ly, a s s u m e t h a t fR; RU j U 2 Asg is a m a t c h e d s ys t e m o f id e m p o t e n t p r e r a d ic a ls
wh ic h s a t is fy t h e c o n d it io n ( P 4 ) a n d ( A; U ) is a m o d u le o f t h e c a t e g o r y M od. Fr o m ( P 4 ) ,
wh ic h is u s e d fo r U -m o d u le RU ( A ) , it fo llo ws t h a t RR(U) ( RU ( A ) ) = RU ( RU ( A ) ) = RU ( A )
b e c a u s e RU is a n id e m p o t e n t p r e r a d ic a l in Mod-U . Th u s , t h e p r o o f o f t h e t h e o r e m is
c o m p le t e d a s R( R ( U ) ) = R( U ) fo r e a c h U 2 As.

De¯nition 6: The radical r of the category Mod is called ideally hereditary if r ( A0; U 0 ) =
r ( A; U ) \ ( A0; U 0 ) for all ideals ( A0; U 0 ) of ( A; U ) 2 Mod [2], [5].

Remar k 2 Th e r a d ic a l R0 o f t h e s ys t e m fR; RU j U 2 Asg is a r a d ic a l o f t h e c a t e g o r y o f
A b e lia n g r o u p Ab. S in c e R0 ( A ) ´ µ R0 ( A) fo r e ve r y A b e lia n g r o u p A a n d e ve r y e n d o m o r -
p h is m ´ o f A, t h e s u b g r o u p R0 ( A) will b e a n U-s u b m o d u le fo r e a c h U -m o d u le A. Th u s , it
is e a s y t o p r o ve t h a t R0 is a r a d ic a l in a ll t h e c a t e g o r ie s Mod ¡ U ( U 2 As) . Th e s ys t e m o f
r a d ic a ls fR; RU j U 2 Asgin wh ic h RU = R0 fo r a ll U 2 As t u r n s t o a p a ir o f fR; R0g. A s
in s u c h s ys t e m s t h e c o n d it io n s ( P 2 ) a n d ( P 3 ) a r e e xe c u t e d a u t o m a t ic a lly, t h e n t h e y will b e
m a t c h e d if t h e y s a t is fy t h e o n ly c o n d it io n : A ¢ R ( U ) µ R0 ( A ) fo r e a c h U-m o d u le A, i.e . t h e
o n ly c o n d it io n ( P 1 ) .

T heor em 4: Assume that r is an ideally hereditary radical in Mod. Then in As it induces
an ideally hereditary radical R and in Ab it induces a torsion RAb such that the pair fR; RAbg
of radicals is matched. Conversely, every matched pair of radicals fR; RAbg, where R is an
ideally hereditary radical in As and RAb is a torsion in Ab, speci¯es a completely determinate
ideally hereditary radical r in Mod whose radical class includes all modules ( A; U ) such that
U = R( U ) and A = RAb ( A ) . There exists a one-to-one correspondence between all ideally
hereditary radicals of Mod and all such matched pairs.

P r oof. L e t r b e a n id e a lly h e r e d it a r y r a d ic a l in Mod a n d le t fR; RU jU 2 Asg b e it s
m a t c h e d s ys t e m o f r a d ic a ls . W e will p r o ve t h a t t h is m a t c h e d s ys t e m , wh ic h in p a r t ic u la r ,
s a t is ¯ e s t h e c o n d it io n ( P 1 ) will b e a p a ir o f fR; R0g, wh e r e R is a n id e a lly h e r e d it a r y r a d ic a l
in As a n d R0 is a t o r s io n in Ab.

L e t U b e a n a s s o c ia t ive r in g a n d A b e a n y U -m o d u le . Co n s id e r t h e m o d u le ( A; 0 ) . It is
e a s y t o p r o ve t h a t ( A; 0 ) is a n id e a l o f m o d u le ( A; U ) . Th e r e fo r e , a s r is a n id e a lly h e r e d it a r y
r a d ic a l in Mod, ( R0 ( A ) ; 0 ) = r ( A; 0 ) = ( A; 0 )

T
r ( A; U ) = ( A

T
RU ( A ) ; 0 ) = ( RU ( A ) ; 0 ) ,

i.e . RU ( A ) = R0 ( A ) fo r e a c h U -m o d u le A, wh e r e U 2 As. Th u s RU = R0 fo r e a c h U 2 As.
N o w le t U b e a n a s s o c ia t ive r in g a n d V b e a n y id e a l o f U. It is o b vio u s t h a t t h e m o d u le

( 0 ; V ) is a n id e a l o f t h e m o d u le ( 0 ; U ) . Th e r e fo r e , a s r is a n id e a lly h e r e d it a r y r a d ic a l in
Mod, ( 0 ; R ( V ) ) = r ( 0 ; V ) = ( 0 ; V )

T
r ( 0 ; U ) ) = ( 0 ; V

T
R ( U ) ) . H e n c e R( V ) = V

T
R ( U ) ,

i.e . R is a n id e a lly h e r e d it a r y r a d ic a l in As.



G. Emin-Teryan (Emin) 9 9

S im ila r ly, c o n s id e r in g t h e m o d u le s ( A; 0 ) , we c a n p r o ve t h a t R0 is a t o r s io n in Ab.
Co n ve r s e ly, le t t h e p a ir o f r a d ic a ls fR; R0g s a t is fy t h e c o n d it io n o f t h e t h e o r e m . A s t h is

p a ir s a t is ¯ e s c o n d it io n ( P 1 ) , it will b e a m a t c h e d s ys t e m o f r a d ic a ls a c c o r d in g t o t h e r e m a r k
a b o ve . Th e r e fo r e , t h e c o n s id e r e d s ys t e m will d e ¯ n e a c o m p le t e ly d e t e r m in a t e r a d ic a l r in
Mod.

N o w s h o w t h a t r is a n id e a lly h e r e d it a r y r a d ic a l. L e t ( A; U ) b e a m o d u le o f M od a n d
( B; V ) b e a n id e a l o f ( A; U ) . S in c e V is a n id e a l o f t h e r in g U , B is a s u b g r o u p o f A b e lia n
g r o u p A, R is a n id e a lly h e r e d it a r y r a d ic a l in As a n d R0 is a t o r s io n in Ab, R ( V ) = V

T
R ( U )

a n d R0 ( B ) = B
T

R0 ( A ) . Th u s r ( B; V ) = ( R0 ( B ) ; R ( V ) ) = ( B
T

R0 ( A ) ; V
T

R( U ) ) =
( B; V )

T
r ( A; U ) .

Th e t h ir d a s s e r t io n o f t h e t h e o r e m fo llo ws fr o m t h e ¯ r s t t wo s in c e t h e e xp r e s s io n s g ive n
t h e r e a r e m u t u a lly in ve r s e .

De¯nition 7: The radical r in the sense of K urosh [1] is called strict if r-radical r ( A; U ) of
any module ( A; U ) contains all r-submodules of ( A; U ) .

De¯nition 8: The radical r of the category Mod is called s t r o n g ly h e r e d it a r y if r ( A1; U1 ) =
r ( A; U )

T
( A1; U1 ) for all submodules ( A1; U1 ) of ( A; U ) 2 M od.

It is e a s y t o p r o ve t h a t e ve r y s t r o n g ly h e r e d it a r y r a d ic a l is a s t r ic t r a d ic a l. In t h e s a m e
wa y we m a y p r o ve t h e fo llo win g t h e o r e m .

T heor em 5: Assume that the strongly hereditary radical r is speci¯ed in Mod. Then in As
it induces a strongly hereditary radical R and in Ab it induces a torsion RAb such that the
pair fR; RAbg of strongly hereditary radicals is matched. Conversely, every matched pair of
strongly hereditary radicals fR; RAbg speci¯es a completely determinate strongly hereditary
radical r in M od whose radical class includes all modules ( A; U ) such that U = R( U ) and A =
RAb ( A) . There exists a one-to-one correspondence between all strongly hereditary radicals of
Mod and all matched pairs of strongly hereditary radicals.

Refer ences

[1 ] G.G. E m in , \ P r e m a n ifo ld s , g r o u p p o id o f m a n ifo ld s a n d s t r ic t r a d ic a ls in t h e c a t e g o r y
o f m o d u le s o ve r a ll r in g s " , Izv. AN Arm. SSR ser. matem., vo l. 1 4 , n o .3 , p p .2 1 1 -2 3 2 ,
1 9 7 9 ; Soviet J ournal of Contemporary M athematical Analysis (Armenian Academy of
Sciences), by Allerton P ress, In c . N e w-Y o r k, p p .5 2 -7 2 , 1 9 8 0 .

[2 ] A . I. K a s h u , R a d ic a ls a n d To r s io n s in t h e m o d u le s , ( in R u s s ia n ) , Iz d . S h t iin z a , K is h in e v,
1 9 8 3 .

[3 ] G. G. E m in -Te r ya n ( E m in ) , \ Id e m p o t e n t p r e r a d ic a ls in t h e c a t e g o r y o f m o d u le s o ve r
a ll r in g s " . P roceedings of CSIT, A r m e n ia , Y e r e va n , p p . 6 1 -6 4 , 2 0 0 5 .

[4 ] G. E m in -Te r ya n ( E m in ) , \ R a d ic a ls in t h e c a t e g o r y o f m o d u le s o ve r a ll r in g s " P roceedings
of CSIT, A r m e n ia , Y e r e va n , p p . 3 5 -3 6 , 2 0 0 7 .

[5 ] G. E m in -Te r ya n ( E m in ) , \ H e r e d it a r y r a d ic a ls in t h e c a t e g o r y o f m o d u le s o ve r a ll r in g s " ,
P roceedings of CSIT, A r m e n ia , Y e r e va n , p p . 4 4 -4 5 , 2 0 1 1 .

Submitted 23.11.2012, accepted 18.02.2013.



1 0 0 Radicals and Preradicals in the Category of Modules over All Rings

è³¹ÇϳÉÝ»ñÁ ¨ ÙÇÝãé³¹ÇϳÉÝ»ñÁ µáÉáñ ûÕ³ÏÝ»ñÇ íñ³ Ùá¹áõÉÝ»ñÇ
ϳﻷáñdzÛáõÙ

¶. ¾ÙÇÝ-î»ñÛ³Ý

²Ù÷á÷áõÙ

¸Çï³ñÏíáõÙ ¿ µáÉáñ ûÕ³ÏÝ»ñÇ íñ³ Ùá¹áõÉÝ»ñÇ M od ϳﻷáñdzÝ: ²Û¹
ϳﻷáñdzÛÇ ûµÛ»ÏïÝ»ñÁ µáÉáñ ³ÛÝåÇëÇ ( A; U ) ½áõÛ·»ñÝ »Ý, áñï»Õ U -Ý ³ëáódzïÇí
ûÕ³Ï ¿ ¨ ³Ýå³ÛÙ³Ý ã¿, áñ áõݻݳ Ùdzíáñ, ÇëÏ A-Ý` ³ç U -Ùá¹áõÉ : ÀݹëÙÇÝ,
( A; U ) Ùá¹áõÉÇó ¹»åÇ ( B; V ) Ùá¹áõÉ ÙáñýǽÙÝ»ñÇ µ³½ÙáõÃÛáõÝÁ µ³Õϳó³Í ¿ ( 'A; 'U )
½áõÛ·»ñÇó, áñï»Õ 'A-Ý ³µ»ÉÛ³Ý A ËÙµÇ ÑáÙáÙáñýǽ٠¿ ¹»åÇ B ËáõÙµ, 'U -Ý U
ûÕ³ÏÇ ÑáÙáÙáñýǽ٠¿ ¹»åÇ V ûÕ³Ï ¨ ( a ¢ u) 'A = a'A ¢ u'U , a 2 A, u 2 U :
²µ»ÉÛ³Ý ËÙµ»ñÇ Ab ϳﻷáñÇ³Ý ¨ ³ëáódzïÇí ûÕ³ÏÝ»ñÇ As ϳﻷáñÇ³Ý Mod-
Ç ÉñÇí »Ýóϳﻷáñdzݻñ »Ý, ÙÇÝã¹»é Mod ¡ U ϳﻷáñÇ³Ý ýÇùëí³Í U ûÕ³ÏÇ
íñ³ ³ç Ùá¹áõÉÝ»ñÇ Ï³ï»·áñÇ³Ý ¿, áñï»Õ U -Ý ó³Ýϳó³Í ³ëáódzïÇí ûÕ³Ï
¿` Mod-Ç »Ýóϳﻷáñdz: ú·ï³·áñÍ»Éáí Mod ϳﻷáñdzÛÇ »ÝóϳﻷáñdzݻñÇ»
”Áëï ß»ñï»ñÇ” Ý»ñϳ۳óÙ³Ý Ñ»ÕÇݳÏÇ ÏáÕÙÇó ³é³ç³ñÏí³Í Ù»Ãá¹Á, Ñݳñ³íáñ
¹³ñÓ³í Ýϳñ³·ñ»É ³Û¹ ϳﻷáñdzÛÇ é³¹ÇϳÉÝ»ñÁ ¨ ÙÇÝãé³¹ÇϳÉÝ»ñÁ: ¸ñ³Ýù
Ýϳñ³·ñí³Í »Ý ³ëáódzïÇí ûÕ³ÏÝ»ñÇ ¨ ýÇùëí³Í ûÕ³ÏÇ íñ³ Ùá¹áõÉÝ»ñÇ
é³¹ÇϳÉÝ»ñÇ, ѳٳå³ï³ë˳ݳµ³ñ, ÙÇÝãé³¹ÇϳÉÝ»ñÇ Ñ³Ù³Ó³ÛÝ»óí³Í, ³ÛëÇÝùÝ,
ѳٳå³ï³ëË³Ý å³ÛÙ³ÝÝ»ñÇÝ µ³í³ñ³ñáÕ Ñ³Ù³Ï³ñ·»ñÇ û·ÝáõÃÛ³Ùµ:

Ðàäèêàëû è ïðåäðàäèêàëû â êàòåãîðèè
ìîäóëåé íàä âñåìè êîëüöàìè

Ã. Ýìèí-Òåðüÿí (Ýìèí)

Àííîòàöèÿ

Ðàññìàòðèâàåòñÿ êàòåãîðèÿ Mod ìîäóëåé íàä âñåìè êîëüöàìè. Îáúåêòû
ýòîé êàòåãîðèè - âñåâîçìîæíûå ïàðû ( A; U ) , ãäå U - àññîöèàòèâíîå êîëüöî, íå
îáÿçàòåëüíî ñ åäèíèöåé, A-ïðàâûé U -ìîäóëü. Ìíîæåñòâî ìîðôèçìîâ ìîäóëÿ
( A; U ) â ìîäóëü ( B; V ) ñîñòîèò èç ïàð ( 'A; 'U ) , ãäå 'A - ãîìîìîðôèçì àáåëåâîé
ãðóïïû A â àáåëåâó ãðóïïó B, à 'U - ãîìîìîðôèçì êîëüöà U â êîëüöî V , ïðè÷åì
( a ¢ u ) 'A = a 'A ¢ u 'U , a 2 A, u 2 U.

Êàòåãîðèÿ àáåëåâûõ ãðóïï Ab è êàòåãîðèÿ àññîöèàòèâíûõ êîëåö As ÿâëÿþòñÿ
ïîëíûìè ïîäêàòåãîðèÿìè êàòåãîðèè M od. Äëÿ ëþáîãî àññîöèàòèâíîãî êîëüöà
U êàòåãîðèÿ M od ¡ U ïðàâûõ ìîäóëåé íàä ôèêñèðîâàííûì êîëüöîì U ÿâëÿåòñÿ
ïîäêàòåãîðèåé êàòåãîðèè M od.

Ñ ïîìîùüþ ïðåäëîæåííîãî àâòîðîì ìåòîäà ”ïîñëîéíîãî” ïðåäñòàâëåíèÿ
èññëåäóåìûõ ïîäêàòåãîðèé êàòåãîðèè M od îïèñàíû ðàäèêàëû è ïðåäðàäèêàëû
ýòîé êàòåãîðèè. Îíè îïèñàíû ñ ïîìîùüþ ñîãëàñîâàííûõ, òî åñòü,
óäîâëåòâîðÿþùèõ íåêîòîðûì óñëîâèÿì ñèñòåì ðàäèêàëîâ, ñîîòâåòñòâåííî,
ïðåäðàäèêàëîâ êàòåãîðèé àññîöèàòèâíûõ êîëåö è ìîäóëåé íàä ôèêñèðîâàííûì
êîëüöîì.