On Annihilator-Extending Modules

: Throughout this work we introduce the notion of Annihilator-closed submodules, and we give some basic properties of this concept. We also introduce a generalization for the Extending modules, namely Annihilator-extending modules. Some fundamental properties are presented as well as we discuss the relation between this concept and some other related concepts.


Introduction:
Throughout this paper we consider that R is a commutative ring with identity and all modules will be unitary left R-modules . It well known that if N has no proper essential extension in M then a submodule N of an R-module M is called closed submodule that is if there exists a submodule K of M with N e K M then N=K [1]. A submodule N of an Rmodule M is called essential submodule of M if for every K M with N K=0 then K=0 [2]. Many authors have been interested in studying the class of closed submodules and some related concepts see [3][4][5]. Yousef and Sahira [6] introduced the notion of Annihilator essential submodules as a generalization of essential submodules where a submodule N of an R-module M is called annihilator-essential if N L=0 then ann(L ) e R .We also modify the thought of annihilator-closed submodules. If N has no proper annihilator-essential extension in M then a submodule N of an R-module M is called annihilator-closed submodule that is if ISSN: 0067-2904

Qasim and Yaseen
Iraqi Journal of Science, 2022, Vol. 63, No. 3, pp: 1178-1183 2211 there exist a submodule K of M such that N a.e K M then N=K. In addition we give some basic properties for this concept. In [7][8][9] authors are interested in study the class of extending modules, and they gave some related concepts, where an R-module M is called Extending module if every submodule of M is essential in a direct summand [10]. In section two we introduce a generalization of extending modules which is called annihilator-extending modules where an R-module M is called annihilator-extending module (shortly by annextending) if every submodule of M is annihilator-essential in a direct summand. We also give some basic properties and we study the relation with the other related concepts.

1.Annihilator closed submodules
We recall the definition of Annihilator essential submodules and some basic properties of this definition are given in lemma(1.2). We introduce the notion of annihilator-closed submodules. We study relation between the closed submodules and the annihilator-closed submodules with some basic properties of this concept. Definition(1.1): [6] A submodule N of an R-module M is called annihilator-essential submodule(shortly by ann-essential) Which is denoted by N a.e M if N L=0 then ann(L) e R for every L M . Lemma(1.2): [6] (1) Every essential submodule is ann-essential submodule. (1) Every ann-closed submodule is closed submodule. Proof: Let N be an ann-closed submodule of M and let K be a submodule of M such that N e K M , by lemma(1.2) every essential submodule is ann-essential then N a.e K M .However N is ann-closed submodule in M so N=K .Thus, N is closed submodule in M.
(2) The following example shows that the converse of (1) in general is not true. Example: Let M=Z 6 as Z-module ,( ̅ ) is closed in Z 6 but not ann-closed submodule since ( ̅ ) is annessential submodule in Z 6 .
(3) It well known that every direct summand is closed submodule [1]. The next example shows that it is not necessary every direct summand is ann-closed submodule Example: : Let M=Z 45 as Z-module , M=( ̅ ) ( ̅ ) both of ( ̅ ) and ( ̅ )are not ann-closed submodule of M since ( ̅ ) a.e M and ( ̅ ) a.e M. Also the Z-module Z 6 =( ̅ ) ( ̅ ) both of ( ̅ )and ( ̅ ) are not annclosed submodules in M . (4) Every module is ann-closed submodule of itself. (5) Since (0) is not ann-essential submodule of any module, then (0) is ann-closed submodule [6]. Proof: Let K be a submodule of M with A B a.e K M then by lemma(1.2) A a.e K M and B a.e K M ,since both of A and B are ann-closed submodules in M then A=K=B hence A B=K. (7) Let M be an R-module and let A be an ann-closed submodule of M . Next example shows that if B is a submodule of M such that A B then it is not necessary that B is ann-closed submodule of M. Example : Let M=Z as Z-module , Z is ann-closed in Z and Z 2Z but 2Z is not ann-closed in Z since 2Z a.e Z . (8) Let A and B be a submodules of an R-module M such that A B M . The following example shows that if B is ann-closed submodule in M then A need not be ann-closed submodule in M. Example :Let M=Z as Z-module, Z is ann-closed submodule in Z and 3Z Z but 3Z is not ann-closed submodule in Z. (9) Let A and B be a submodules of an R-module M such that A B M . The following example shows that if A is ann-closed submodule in M then B need not be ann-closed submodule in M. Example: Let M=Z as Z-module and the submodules A=(0) and B=2Z . Note that A is annclosed submodule in Z but B is not ann-closed submodule in Z since 2Z a.e Z.

2.Annihilator-Extending Modules.
In this section we introduce the concept of annihilator-extending modules as a generalization of extending modules and we also give some basic properties for it. Definition(2.1): An R-module M is called annihilator extending module (shortly annextending module) if every submodule of M is ann-essential in a direct summand . A ring R is called ann-extending ring if every ideal of R is ann-essential in direct summand .

Remarks and Examples(2.2):
(1) The Z-module Z 6 is ann-extending module since every submodule is ann-essential in a direct summand .
Proof: Since every uniform module is an extending module [8] then the result follows directly from (2) .Now, assume that M is decomposable module then there exist two non zero submodules N and K such that M=N K so both of N and K are not essential submodules which is a contradiction since M is uniform module . The converse in general is not true for example : consider the Z-module Z 6 is ann-extending module since every submodule is annessential in direct summand but Z 6 is not uniform module. Recall that: an R-module M is called annihilator-uniform(shortly by ann-uniform) module if every submodule of M is ann-essential submodule [6]. (5) Every ann-uniform module is an ann-extending module. Proof: Let N be a submodule of M .If N=(0) then clearly that N is ann-essential in (0) which is a direct summand of M . If N (0) ,since M is ann-uniform module then N a.e M, but M is direct summand of it self ,so M is ann-extending module . Next example shows that the converse in general is not true. Example: Let M= Z 24 as Z 24 module , M is ann-extending but not ann-uniform module. In similar way we can prove that A B is a direct summand of B .
Proposition(2.6); Let M be a finitely generated faithful multiplication R-module .If R is an ann-extending module then M is an ann-extending module. Proof: Let N be an ann-closed submodule in M ,since M is multiplication R-module then N=(N: R M)M [11] .since N a.c M then by proposition(1.12) (N: R M) a.c R ,but R is annextending module hence ( ) We assume that M is an ann-extending module , and let N M ,by hypothesis there exist an ann-closed submodule H of M such that N e H ,since M is an ann-extending module then H is a direct summand of M and hence M is an extending module.