More on Stability and Dual Stability of Submodules

More on Stability and Dual Stability of Submodules

Asaad M. A. Al-hosainy

Ass. Prof. in Mathematics, College of Educations for Sciences University of Babylon, Iraq

Abstract- The aim of this paper is to introduce an indicator for d-stability of any submodule in a module. If N is a submodule of a module M, the intersection of all kernels of the homomorphisms of M into the quotient of M by N is

results and properties due to d-stability. This indicator, also, will help to give shorter proofs for old results.

Let N be a submodule of a module M. The d-stability indicator of N in M, denoted by k M (N), is defined by

considered. Dually, the sum of all images of the

homomorphisms from a submodule N into the module M, as an indicator for the stability of N in the module M, is

k M ()

ker .

Hom(, )

considered, too.

Keywords: d-stable; fully d-stable; quasi-projective; stable; fully stable; quasi-injective.

1. INTRODUCTION

   Throughout this paper, modules are left unitary modules over an associative ring with identity.

   A submodule N of a module M is said to be dual stable(shortly d-stable) if N ker for each

   Hom(M, M/N ) [2]. A module is fully d-stable if all

   its submodules are d-stable [2]. These concepts were introduced and studied widely by M. Abbas and the author in many previous papers (see, [2], [3], [4], and [5]). The class of fully d-stable modules is a subclass of duo modules, and it is contained in the class of multiplication modules (the general version, over rings not necessary commutative, see [7]). In fact it lies strictly between those two classes ([2] and [5]). The intersection of the class of quasi-projective with the class of duo modules is a subclass of fully d-stable modules[2]. In certain conditions, the concepts, full d-stability and quasi-projectivity coincide[5].

   The cases in which fully d-stable modules become multiplication, also, were discussed in [5]. So the importance of fully d-stable modules can be seen in view of its position among duo, multiplication and quasi- projective modules. In addition, full d-stability has many generalizations ( see [3] and [4]), which react with known concepts, in particular "full pseudo d-stability coincide with a dual to the concept of terse modules (due to Weakley [8]), see [4].

   In this paper an indicator for the d-stability of a submodule will be introduced, which will tend to new

   In section 2, the properties of k M (N), and its relation

   with the d-stability of N, will be discussed. Many new

   results about d-stability, with the help of k M (N), also proved. As a sample, the intersection of any family of d- stable submodules is again d-stable. While in the case of sum, a condition of being quasi-projective (the module) is needed.

   The dual of the new concept, will be studied in section

   3. M. Abbas in1991 introduced the concept of stable submodule N in a module M. A submodule N of a module M is said to be stable if, (N) N for each

   Hom(N, M) [1]. The indicator of stability of N,

   denoted by I M (N) , will be defined in this paper by I M (N) = Im .

   Hom( N, M)

   As in section 2, the indicator of stability has many properties, and serves to prove new results related to stability. The sum of any family of stable submodules is again stable. The intersection of stable submodules in a quasi-injective module is stable, too.

2. INDICATOR OF D-STABILITY

1. Definition.

   Let be a submodule of a module M , we define

   k M () ker .

   Hom(,)

2. Proposition.

   Let be a submodule of a module M , then

   k M (N) is a submodule of M and (i) k M () .

   1. is d-stable if and only if k M () .

   2. M is fully d-stable if and only if k M () for each submodule of M .

   Proof: Immediate by definitions.

3. Lemma

Let M be a module , and L be submodules of

M such that L k M (N) , then

L),(ML(NL)) , where is an isomorphism from M/N onto (M/L)/(N/L). By d-stability of N/L in M/L, N/L ker( ) = ker which implies

(by definition of ) (N)= 0, that is N ker . Therefore N is d-stable in M.

It was proved in [2] , that " If M is fully d-stable and

is a submodule of M , then is also fully d-

k M L k(M L) L) .

stable". This result, now, can be concluded as a corollary of Proposition (D).

Proof: Let homomorphism,

: L L)(NL) be any

: M M L be the natural

More precisely, Proposition (D) gives the following result.

epimorphism, and : (M L) (N L) M N be an

isomorphism. Then an easy check shows that

ker ker ( ) L , that is, for each

F. Corollary.

If is a submodule of a module M such that any

containing submodule is d-stable in M . Then is

Hom((M L),(M L (N L)) , there is a

fully d-stable.

Hom(M, M N) such that ker ker L .

This implies, k M L k(M L) L) .

D. Proposition.

If is a d-stable submodule of a module M and L

N , then N L is d-stable in M L .

1. Lemma.

   If M is a module and f End(M) , then

   f (k M (N)) N .

   Proof: Let f End () , be the natural

   Proof: is d-stable implies k M ()

   epimorphism of M onto k M () , and f ,

   (B,(ii)), hence L k M (N) , then by (C)

   let : k M () be defined by

   N L k

   (M L)

   (N L) . Therefore by (B, (i) and (ii))

   (x k M () x . is well defined

   NL is d-stable in L .

   The converse of the above proposition is not true, that is, if N/L is d-stable in M/L, it is not necessary that N be d-stable in M. Note the following situation:

   Let A be a fully d-stable module, M= A A, L= A 0 and N= A B, where B is a submodule of A, then N/L is d-stable in M/L while N is note d-stable in M ( in fact N is even not fully invariant in M).

   But the converse of Proposition D can be proved if we

   homomorphism since k M () , also

   f : , so by definition of

   k M () ,

   M

   M

   M

   M

   k () ker( f ) f 1 (1 (ker()) , but ker k () and 1 k ()) , hence k () f 1 () which implies f (k M ()) .

2. Lemma

add certain condition.

E. Theorem.

For each f End () , the induced

Let M be a module, L N M. If N/L is d-stable in M/L and L k M () , then N is d-stable in M.

Proof: Let Hom(M, M/N ), by the condition L k M () , we can define a homomorphism : M/L M/N, by (x+ L) = (x). Then

map f : k() , defined by

f (x k()) f (x) is a well defined

homomorphism.

Proof: Clear by Lemma G.

I. Proposition. Proof: (i) clear by (I) and the above remark.

Let be a submodule of a module M . If M is

(ii) Let k M () L , with f (L) for each

quasi-projective then k M (N) is d-stable in M .

f End () . If Hom(, )

and

f End ()

be such that

f

, where is

Proof: Let Hom(, k M ()) , and assume

the natural epimorphism of M onto

, ( M is

that

k M () ker() , then there exists x k M ()

with (x) y k M () and y k M () , hence

quasi-projective module), then

(L) ( f (L)) 0 ,

( y) 0 for some

Hom(, ) . Since M

that is, L ker for each

Hom(, ) ,

is quasi-projective, there exists

f , g End () with

hence L k M () .

f , g where and are the natural

M. Theorem.

epimorphisms of M onto k M () and

The intersection of any family of d-stable submodules of a module is again d-stable.

respectively. Define : k M () by

(t k M () g(t) , is well defined by (2.8)

Proof: Let

{Ni }iI

be a family of d-stable

and Hom(, ) , hence ( )(x) 0

submodules of a module M , N Ni

iI

and let

(since

x k() ) , but

Hom(M, M N) . For each

i I , define

((x)) (y k M ()) g(y) (y) 0 ,

i N, Ni ) by i (x N) x Ni ,

a contradiction.

then i are well-defined since

N Ni . Let

J. Corollary.

i i ,

i I . Since Ni are d-stable, it follows

i

i

If M is a quasi-projective module, then

Ni ker i

1 (ker ) 1 (N

N) . So

k M (k M ()) k M ()

for any submodule of

(Ni ) Ni N

for each

i I . Now

M .

(N) (Ni ) (Ni

N) Ni

N 0 .

iI

iI

iI

In [2], it was proved that that a quasi-projective module is fully d-stable if and only if it is duo . A more general result can be stated in the following, the proof is as in [2] .

K . Remark.

If M is a quasi-projective module, and is a submodule of M , then is d-stable if and only if it is fully invariant in M .

Using the above remark and Proposition I, the following can be added.

L. Corollary.

If M is quasi-projective module, a submodule

This means N ker , and hence is d-stable in

M .

In [2], a lemma was proved to serve finding an example of a fully d-stable module which is not quasi- projective. The lemma says " If is an R -module having exactly three nontrivial submodules, 1 , 2

and 1 2 , with 1 not isomorphic to

2 , then is a fully d-stable module which is not quasi-projective." ( Lemma 3.7, [ 2]). Now a shorter proof for the first part of this lemma (the proof of full d- stability) can be given in the following .

Proof: By the hypothesis 1 and 2 are maximal

of M , then

submodules with

1

not isomorphic to

2 ,

1. k M () is fully invariant in M .

2. k M () is the largest submodule L of M

hence by ( a note after Example 3.5, [3]) both

1 and 2 are d-stable. Then by (Theorem M),

contained in with the property f (L) for each

f End () .

1 2 is d-stable too. Hence is fully d-stable.

The sum of d-stable submodules of a module need not be d-stable, as it is seen in the following example.

Proof: Let

{N j}jJ

be a family of d-stable

submodules of a module M , then by Remark K each

1. Example.

   Referring to an example of Hallett [6], where R is an

   j is fully invariant. If

   f End () , then

   algebra over

   2

   having basis

   f ( j ) j , j J , hence

   { e , e , e , n , n , n , n

   }with the following

   f ( j ) f ( j ) j

   , that is , N j

   1 2 3 1 2 3 4

   multiplication table:

   jJ

   jJ

   jJ

   jJ

   is fully invariant, and (again by (K)) it is d-stable in M .

   	e1	e2	e3	n1	n2	n3	n4
   e1	e1	0	0	n1	n2	0	0
   e2	0	e2	0	0	0	0	0
   e3	0	0	e3	0	0	n3	n4
   n1	0	n1	0	0	0	0	0
   n2	0	0	n2	0	0	0	0
   n3	n3	0	0	0	0	0	0
   n4	0	n4	0	0	0	0	0

   P. Theorem.

   If M is a quasi-projective module,

   , K

   are

   M

   submodules of

   submodules of M then

   and

   i iI

   is a family of

   1. If then k M

      () k M

      () .

   2. k M () is the largest d-stable submodule of M

      contained in .

      Let M Re1 Rn 2 , then M has the following

   3. k M (i ) k M (i )

      and

      submodules:

      iI

      iI

      N Re1

      , K Rn 2

      Rn3

      , I Rn3

      , and

      k M (i ) k M (i ) .

      J Rn 2 . The lattice of these submodules is

      iI

   4. If L

      iI

      is the family of all d-stable submodules

      j jJ

      M of M contained in then k M () L j .

      jJ

      Proof: (i) Let

      N K

      and

      x k M (K) , then there

      exists

      Hom(, )

      with

      (x) 0 . Since

      I J M is quasi-projective, there exists

      f End () such

      that (x) f (x) , hence

      f (x) K . Let

      0 Hom(, )

      defined by

      f , where

      Note that N and K are maximal in M , with

      is the natural epimorphism of M onto , then

      MN K , hence N and K both are not d-stable (see Corollary 3.4, [3]). While I and J are d-stable submodules of M (easy check), and I+J=K which is not d-stable.

      (x) f (x) 0 ( since ) , hence x ker , and so, x k M () . Therefore k M () k M () .

      With quasi-projectivity , d-stability will be closed

      1. Assume that

         k M () L , where L is d-

         under sum of submodules as is shown in the following.

         stable, then by (i)

         k M (L) k M () , but

2. Theorem.

If M is a quasi-projective module, then the sum of any family of d-stable submodules of M is again a d- stable submodule of M .

k M (L) L , since L is d-stable, hence L k M () , and so. L k M () .

1. By (i) k M ( i ) k M (i ) for each i I ,

   iI

   Hence k M (i ) k M (i ) . On the other hand

   1. Proposition.

      If are submodules of a module M , and if

      is stable in M , then it is stable in .

      iI iI

      k M (i ) is a d-stable submodule contained in

      iI

      i (Proposition I and Theorem M) , hence

      iI

      k ( ) k ( ) (by (ii)), therefore

      Proof: Clear by Lemma C and Proposition B, (ii).

   2. Remark.

      If x is an element of a module M over a ring R , then IM (Rx) will be simply denoted by IM (x) . It is

      M i

      iI

      M i

      iI

      clear by Proposition(3.2) and (Corollary 1.5 in [1]) that :

      k M (i ) k M

      (i ) .

      M is fully stable if and only if

      1. in M .

         IM (x) = Rx for each

         iI

         Also by (i)

         iI

         k M (i ) k M (i ) , which implies

         iJ

   3. Proposition.

   k M iI

   (i ) k M

   (i ) .

   iI

   If is a submodule of a module M , then IM () is stable in M .

2. Since k M () is a d-stable submodule contained in

Proof: For simplicity denote I

(N)

by . Let

, k

() L

and so k

() L

M

, on

M j jJ

M j

jJ

: be any homomorphism, then

the other hand , by(ii) and Proposition(2.15),

Hom(, ) ,for each homomorphism

L j k M () . therefore , k M () L j .

: , since () . Now,

jJ

jJ

() (

Im)

Hom(,)

(Im)

Hom(,)

III. THE INDICATOR OF STABILITY

Im()

Hom(,)

Im

Hom(,)

1. Definition.

   Let be a submodule of a module M , we define

   Therefore, is stable in M .

   Remark. If f End() , and if J is a stable

   IM ()

   Im .

   Hom( N,M)

   submodule containing a submodule of M ,then

   f () J . (clear)

2. Proposition.

Let be a submodule of a module M , then

G. Corollary.

IM (N) is a submodule of M and (i) N IM () .

1. is stable if and only if IM () .

   If is a submodule of a quasi-injective module

   M , then IM (N) is the smallest stable submodule of

2. M is fully stable if and only if IM () for each submodule of M .

Proof: Immediate by definitions.

M containing .

Proof: Denote IM (N)

submodule containing .

n

by . Let J be any stable

C. Lemma.

y

implies

y i (xi ), i Hom(, ) ,

i1

If M is a module and are submodules of

M , then () () .

1. and n a positive integer.

Let f i be an extension of i to M for

i 1,2,…, n ,

Proof: It is clear since it can be considered that

then i (xi ) fi (xi ) J

for i 1,2,…, n ( since

Hom(, ) Hom(, ) .

xi J , and

f () J by the above remark).

Therefore y and then J .

In [1], it was mentioned that the sum of any family of stable submodules of a module is again a stable

IM ( j )

jJ

is a stable submodule, also it contains

submodule. In the following, the intersection will be

j , this implies IM ( j ) IM ( j ) . On

discussed .

jJ

jJ

jJ

H. Theorem.

the other hand ( j ) ( j ) for each

jJ

j J ,

If M is a quasi-injective module, and i iI

is a

since j j and by (i). Hence

family of stable submodules of M , then Ni is

i

jJ

( ( j )) ( j ) , therefore

stable in M .

jJ

jJ

IM ( j ) IM ( j ) .

Proof: Let : i , f End() be an

jJ

jJ

i

extension of ( M is quasi-injective), and let

i be

(iii) It is clear that

I () L j jJ

(by Corollary G)

the restriction of f to i for each i , then

(i ) (i ) i (i ) i .

and (ii), since L j ).

jJ

i

i

i

i

On the other hand

I () is a member of the family

Corollary G and Theorem H will help to prove the following properties of IM (N) .

I. Theorem.

If M is a quasi-injective module, , K are

L j jJ , hence L j () .

jJ

submodules of M and is a family of

j jJ

submodules of M then

1. If then IM () IM () .

2. IM ( j ) IM ( j )

   and

   jJ jJ

   IM ( j ) IM ( j ) .

   jJ

3. If L

jJ

is the family of all d-stable submodules

j jJ

of M containing then I () L j .

jJ

Proof: (i) If

, then by Proposition B ,(i) ,

() . But IM (N) is the smallest stable

submodule containing ( Corollary E), hence

IM () IM () .

(ii) By Theorem H,

IM ( j )

jJ

is stable, also

j IM ( j ) , so IM ( j ) IM ( j )

jJ

jJ

jJ

jJ

by Corollary G. For the sum, by the note after G,

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