On Finitely Annihilated Modules

Let R be a commutative ring with identity and M be a unitary R-module. An R-module M is called finitely annihilated if there exists a finitely generated R-submodule N of M such that ann(M)=ann(N).Our main purpose in this work is to study this property in some known classes of modules such as quasi-injective, multiplication and other modules. We prove that: 1-If M is a quasi-injective R-module, then M is finitely annihilated if and only if M is finendo. 2-If M is a multiplication R-module, then M is finitely annihilated if and only if M is finitely generated. 3-M is a faithful finitely annihilated R-module if and only if M is a compactly faithful R-module. Introduction Let R be a commutative ring with unity and let M be a unitary R-module .C. Faith called an R-module M is bounded if there exist an element x M  such that ann(M)=ann(x)(Faith,1970), and he studied some properties of these modules. Also other properties of bounded modules were studied in(Ameen,1992).John A. Beachy and William D.Blair gave a generalization to the bounded module concept (Beachy&Blair,1978). They called an R-module M is finitely annihilated if there exists a finite set  n x x x ,..., , 2 1 ,Where , 1,2, , i x M i n    , such that     1 2 ( ) , , , n ann M ann x x x  . It is clear that every bounded R-module is finitely annihilated. In section one, we study some properties of finitely annihilated R-modules. In section two, we present sufficient conditions for quasi-injective modules to be finitely annihilated and we study some properties of quasi-injective R-modules satisfy finitely annihilated property. Journal of Kirkuk University – Scientific Studies , vol.3, No.2,2008 We prove that if M is a finitely annihilated quasi-injective R-module, then M is finitely generated over End(M)(Th.2.1). Also we prove that if M is a quasi-injective R-module, then M is finitely annihilated if and only if M is finendo(Corollary 2.2). Section three is devoted to study finitely annihilated property in the class of multiplication R-modules. We look for necessary or sufficient conditions for multiplication R-modules to be a finitely annihilated R-module. We prove that if M is multiplication Rmodule, then M is finitely annihilated if and only if M is finitely generated(Prop.3.1).In section four , we study finitely annihilated property in other classes of modules such as quasi-Dedekind, compressible, Fregular and compactly faithful. Some Basic Properties of Finitely Annihilated modules. In (Beachy&Blair, 1978), an R-module M is called finitely annihilated if there exists a finite set   1 2 , , n x x x in M such that,       1 2 , , n ann M ann x x x  . In this section, we present an equivalent statement for this concept. Furthermore, we study some properties and give a characterization for this concept. The proof of the following remark is easy and hence is omitted. Remark1.1: Let M be an R-module. M is finitely annihilated if and only if     ann M ann N  for some finitely generated R-submodule N of M. Examples and Remarks 1.2: 1-Every torsion free R-module, where R is an integral domain, is finitely annihilated. 2-Every finitely generated R-module is finitely annihilated. But the converse is not true. For example Q (the set of all rational numbers) as a Z-module is finitely annihilated but not finitely generated, 3p Z  as a Z-module is not finitely annihilated. 4-The homomorphic image of finitely annihilated R-module may not be finitely annihilated. For example a Z-module Q is finitely annihilated but Q Z is not finitely annihilated Z-module. 5The direct summand of finitely annihilated R-module may not be finitely annihilated. For example, the Z-module p M Z Z    is finitely annihilated Journal of Kirkuk University – Scientific Studies , vol.3, No.2,2008 but p Z  is not finitely annihilated. The proof of the following proposition is straightforward and hence is omitted. Proposition 1.3: Let M be an R-module and let I be an ideal of R such that ( ) I ann M  . Then M is finitely annihilated R-module if and only if M is finitely annihilated R I -module. The following result is an immediate consequences of Prop.1.3. Corollary 1.4: Let M be an R-module. Then M is a finitely annihilated R-module if and only if M is a finitely annihilated ) (M ann R -module Proposition 1.5:Let 1 M and 2 M be two finitely annihilated R-modules. Then 1 2 M M  is a finitely annihilated R-module. Proof: Since 1 M is finitely annihilated, there exists a finitely generated R-submodule 1 N of 1 M such that     1 1 ann M ann N  . Similarly, there exists a finitely generated R-submodule 2 N of 2 M such that     2 2 ann M ann N  . It is clear that     1 2 1 2 ann M M ann N N    . Now, let   1 2 r ann N N   , then     , 0,0 r x y  for all 1 x N  and for all 2 y N  , that is     , 0,0 rx ry  . Hence 0 rx  , for all 1 x N  , and 0 ry  , for all 2 y N  . This implies that         1 2 1 2 r ann N ann N ann M ann M     . Whence   1 2 r ann M M   . This prove that     1 2 1 2 ann N N ann M M    . Therefore     1 2 1 2 ann M M ann N N    , proving that 1 2 M M  is finitely annihilated. The following result is an immediate consequence of Prop.1.5. Corollary 1.6: A finite direct sum of finitely annihilated R-modules is finitely annihilated. However, an infinite direct sum of finitely annihilated Rmodules may not be a finitely annihilated R-module, as it is shown in the following example. Journal of Kirkuk University – Scientific Studies , vol.3, No.2,2008 Example 1.7: p Z as a Z-module is finitely annihilated for all prime p by 1.2(2), but p Z  is not a finitely annihilated Z-module. The following characterization is appeared in (Beachy&Blair, 1978) Proposition 1.8: M is finitely annihilated R-module if and only if   R ann M is embedded in k M , where k is a positive integer. The following remark is needed in the proof of next proposition. Remark 1.9: Let M be an R-module. If N is E-submodule of M , where E= End(M), then N is R-module of M. Proof: It is easy. The following result is a consequence of Remark 1.9. Proposition 1.10: Let M be an R-module. If M is a finitely annihilated E-module, where E= End(M), then M is a finitely annihilated R-module. Proof: Since M is finitely annihilated E-module, then there exists a finitely generated E-submodule N of M such that     E E ann M ann N  .Let   1 2 , , s x x x be a set of generator of N, where , 1,2, , i x N i s   .Thus     1 2 , , s N x x x  . Let K be an Rsubmodule of M generated by   1 2 , , s x x x . We claim that     R R ann M ann K  . Let   r ann K  , and define : f M M  such that f(m)=rm, for all m in M. Thus   0, 1,2, i i f x rx i s     .By Remark 1.9, N is an R-submodule of M. Let n  N,then       1 1 2 2 s s n h x h x h x     , where , 1,2, , i h E i s    . Hence               1 1 2 2 1 2 0 0 0 0 s s s f n rn h rx h rx h rx h h h           and consequently f (N) =rN=0. Therefore   E f ann N  .But     E E ann M ann N  so that   E f ann M  . This means that f(M)=rM=0. Whence ( ) r ann M  . It is clear that     ann M ann K  .◘ Recall that an R-module M is said to be finendo if it is finitely generated over End(M) (Faith,1970). Corollary 1.11: Let M be an R-module. If M is finendo, then M is finitely annihilated. Journal of Kirkuk University – Scientific Studies , vol.3, No.2,2008 Proof: Since M is finendo, thus M is finitely generated over End(M).By 1.2(2),M is finitely annihilated E-module. Thus by Prop.1.10, M is finitely annihilated R-module. In the following proposition, we investigate the behavior of finitely annihilated property under localization. Proposition 1.12: If M is a finitely annihilated Rmodule, then S M is a finitely annihilated S R -module, where S is a multiplicatively closed set of R. Proof: Suppose that M is a finitely annihilated R-module, then there exists a finitely generated Rsubmodule N of M such that     ann M ann N  . Since N is finitely generated, then S N is finitely generated and       S R S S ann N ann N  .It is clear that ann RS (M S )ann RS (N S ). Let       S R S S r ann N ann N s   . Thus     r ann N ann M   and t  S. Let , , , S m x M x m M s S s     .Whence 0 r m rm t s ts ts    . This implies that ( ) S R S r ann M t  . Therefore     S S R S R S ann M ann N  which proves that S M is a finitely annihilated S R -module. Finitely Annihilated Modules and Quasi-Injective Modules. An R-module M is said to be quasi-injective if for each R-submodule N of M and every R-homomorphism from N to M can be extended to an Rendomorphism of M (Faith, 1970). In this section, we look for conditions for quasi-injective modules to be finitely annihilated. We begin with the following theorem which gives a condition under which the converse of Corollary 1.11 is true. Theorem2.1: If M is a quasiinjective finitely annihilated R-module, then M is finendo. Proof: By prop. 1.8, 0 ( ) g k R M ann M   is exact. We claim that   ( , ) , , 0 ( ) R Hom g I k R R R Hom M M Hom M ann M         is exact. Journal of Kirkuk University – Scientific Studies , vol.3, No.2,2008 Let , ( ) R R Hom M ann M         . Since g is monomorphism, then g( ) (M ann R )LM k ,,where L is an Rsubmodule of k M . Let 1 1 : ( ) g j k R j g L M M ann M          where j is the injection homomorphism. Consider the following diagram where i is the inclusion mapping. Since M is quasi-injective, then k M is quasi-injective (Faith, 1970).Thus there exists a homomorphism : k k M M   such that i    . That is L    . Let : k k M M M         , where : k M M   be the canonical projection. Thus   1 , Hom g I I g I g I g I j g g I j I        


Introduction
Let R be a commutative ring with unity and let M be a unitary R-module .C. Faith called an R-module M is bounded if there exist an element xM  such that ann(M)=ann(x) (Faith,1970), and he studied some properties of these modules. Also other properties of bounded modules were studied in (Ameen,1992).John A. Beachy and William D.Blair gave a generalization to the bounded module concept (Beachy&Blair,1978 . It is clear that every bounded R-module is finitely annihilated. In section one, we study some properties of finitely annihilated R-modules. In section two, we present sufficient conditions for quasi-injective modules to be finitely annihilated and we study some properties of quasi-injective R-modules satisfy finitely annihilated property. We prove that if M is a finitely annihilated quasi-injective R-module, then M is finitely generated over End(M)(Th.2.1). Also we prove that if M is a quasi-injective R-module, then M is finitely annihilated if and only if M is finendo(Corollary 2.2). Section three is devoted to study finitely annihilated property in the class of multiplication R-modules. We look for necessary or sufficient conditions for multiplication R-modules to be a finitely annihilated R-module. We prove that if M is multiplication Rmodule, then M is finitely annihilated if and only if M is finitely generated(Prop.3.1).In section four , we study finitely annihilated property in other classes of modules such as quasi-Dedekind, compressible, Fregular and compactly faithful.

Some Basic Properties of Finitely Annihilated modules.
In (Beachy&Blair, 1978) . In this section, we present an equivalent statement for this concept. Furthermore, we study some properties and give a characterization for this concept. The proof of the following remark is easy and hence is omitted.

Remark1.1:
Let M be an R-module. M is finitely annihilated if and only if     ann M ann N  for some finitely generated R-submodule N of M. Examples and Remarks 1.2: 1-Every torsion free R-module, where R is an integral domain, is finitely annihilated. 2-Every finitely generated R-module is finitely annihilated. But the converse is not true. For example Q (the set of all rational numbers) as a Z-module is finitely annihilated but not finitely generated, 3p Z  as a Z-module is not finitely annihilated. 4-The homomorphic image of finitely annihilated R-module may not be finitely annihilated. For example a Z-module Q is finitely annihilated but Q Z is not finitely annihilated Z-module. A finite direct sum of finitely annihilated R-modules is finitely annihilated. However, an infinite direct sum of finitely annihilated Rmodules may not be a finitely annihilated R-module, as it is shown in the following example.

Proof:
It is easy. The following result is a consequence of Remark 1.9. Proposition 1.10: Let M be an R-module. If M is a finitely annihilated E-module, where E= End(M), then M is a finitely annihilated R-module.

Proof:
Since M is finitely annihilated E-module, then there exists a finitely generated E-submodule N of M such that • Recall that an R-module M is said to be finendo if it is finitely generated over End(M) (Faith,1970).

Corollary 1.11:
Let M be an R-module. If M is finendo, then M is finitely annihilated.

Proof:
Since M is finendo, thus M is finitely generated over End(M).By 1.2(2),M is finitely annihilated E-module. Thus by Prop.1.10, M is finitely annihilated R-module. In the following proposition, we investigate the behavior of finitely annihilated property under localization.

Proposition 1.12:
If M is a finitely annihilated R-module, then S M is a finitely annihilated S R -module, where S is a multiplicatively closed set of R.

Proof:
Suppose that M is a finitely annihilated R-module, then there exists a finitely generated R-submodule N of M such that

Finitely Annihilated Modules and Quasi-Injective Modules.
An R-module M is said to be quasi-injective if for each R-submodule N of M and every R-homomorphism from N to M can be extended to an Rendomorphism of M (Faith, 1970). In this section, we look for conditions for quasi-injective modules to be finitely annihilated. We begin with the following theorem which gives a condition under which the converse of Corollary 1.11 is true.

Theorem2.1:
If M is a quasi-injective finitely annihilated R-module, then M is finendo. Proof: where j is the injection homomorphism. Consider the following diagram where i is the inclusion mapping. Since M is quasi-injective, then k M is quasi-injective (Faith, 1970 Let M be Q-module. Then the following statements are equivalence: 1-Every R-submodule of M is finitely annihilated. 2-Every R-submodule of M is finendo. 3-Every R-submodule of M is finitely generated over End(M). Proof: (1)  (2) suppose that every R-submodule of M is finitely annihilated. Let N be an R-submodule of M. Thus N is a quasi-injective finitely annihilated R-submodule. By Th.2.1, N is finendo. x L  such that for each yL  , . Let X be a finitely generated R-submodule of M generated by 12 , , , n x x x .It is clear that XL  .We claim that ann(L)=ann(X  (Abass, 1990).

Proposition 2.6:
Let M be a fully stable quasi-injective R-module. Then M is finitely annihilated if and only if M is finitely generated.

Proof:
If M is finitely annihilated, then there exists a finitely generated Rsubmodule N of M such that ann(M) =ann(N).By (Abass,1990) Let M be a fully stable semi-simple R-module. Then M is finitely annihilated if and only if M is finitely generated.

Finitely Annihilated Property with Multiplication Modules.
An R-module M is said to be multiplication module if for every Rsubmodule N of M, there exists an idea I in R such that N=IM (Barnard,1981) .The following proposition shows that the converse of 1.2(2) is true in the class of multiplication modules.

Proposition 3.1:
Let M be a multiplication R-module. M is finitely annihilated if and only if M is finitely generated.

Proof:
Since M is a finitely annihilated R-module, then there exists a finitely generated R-submodule N of M such that Ann (M)=ann(N). By (Low&Smith, 1990), M is finitely generated. The converse follows from 1.2(2).The condition multiplication in Prop.3.1 can not be dropped. For example Q as Z-module is finitely annihilated, but Q is not finitely generated and not multiplication.

Corollary 3.2:
If M is a finitely annihilated multiplication R-module, then

Proof:
It follows from Prop.3.1 and (Naoum, 1990).In the class of multiplication modules the converse of Prop.1.10 is true as the following proposition indicate that.  . This completes the proof that M is finitely annihilated. Recall that an R-submodule N of an R-module M is dense in M if * , ( ) 0 f M f N    , then f=0 (Naoum, 1990).

Proposition 3.5:
If M is a torsionless multiplication R-module and contains a finitely generated dense R-submodule N, then M is finitely annihilated.

Proof:
It is clear that If M is a multiplication R-module and contains a finitely generated essential R-submodule N with Z(N)=0, then M is finitely annihilated.

Proof:
It is enough to prove that M is non-singular, that is ( ) 0 ZM  . Suppose that ( ) 0 ZM  , then there exists a nonzero element mM  such that ann(m) is essential in R. But N is essential in M, then there exist r  R such that 0 rm N  . Since ( ) ( ) ann m ann rm  , so () ann rm is essential in R. Thus ( ) 0 rm Z N  . Therefore rm=0 which is a contradiction. Hence ( ) 0 ZM  , that is , M is non-singular. By Prop.3.6, M is finitely annihilated. •.

Corollary3.8:
Let R be a ring such that ( ) 0 ZR . If M is multiplication torsionless Rmodule which contains a finitely generated essential R-submodule, then M is finitely annihilated.

Finitely Annihilated Property with Some Types of Modules.
In this section, we study the relationship between finitely annihilated modules and other modules. An R-module M is said to be prime if ann M ann N  for every non-zero submodule N of M (Beachy, 1976). It is clear that every prim R-module is finitely annihilated, but the converse is not true in general as the following example shows.  (Mijbass, 1997). It is known that every quasi-Dedekind module is prime (Mijbass, 1997) , and as an immediate consequence of this result the following proposition.

Proposition4.1:
If M is a quasi-Dedekind R-module, then M is finitely annihilated. Recall that an R-module is said to be Dedekind module if every submodule of it is invertible and an R-module is called prufer module if every finitely generated submodule of it is invertible (Al-Alwan,1993). In particular, every Dedekind module and every prufer module is quasi-Dedekind (Mijbass,1997) . We get the following corollaries that are direct results from the Prop.4.1

Corollary 4.2:
if M is Dedekind R-module, then M is finitely annihilated module.

Corollary 4.3:
If M is prufer R-module, then M is finitely annihilated module. An R-module is called compressible if every non-zero submodule of M contains an isomorphic copy of M (Desale & Nicholoson, 1981).

Proof:
Suppose that M is faithful finitely annihilated. By Prop.1.8, 0 k RM  is exact for some integer k > 0. Thus M is compactly faithfulR-module. Conversely; Since M is compactly faithful , so 0 n RM  is exact for some integer n > 0 ,and hence M is faithful. By Prop.1.8, M is finitely annihilated.

Corollary4.12:
If M is a finitely annihilated semi-simple R-module, then