T-Stable-extending Modules and Strongly T- stable Extending Modules

In this paper we introduce the notions of t-stable extending and strongly t-stable extending modules. We investigate properties and characterizations of each of these concepts. It is shown that a direct sum of t-stable extending modules is t-stable extending while with certain conditions a direct sum of strongly t-stable extending is strongly t-stable extending. Also, it is proved that under certain condition, a stable submodule of t-stable extending (strongly t-stable extending) inherits the property.


Introduction
Let be a ring with unity and be a right -module. A submodule of is called essential in ( if , implies A submodule of is called closed in if it has no proper essential extension in , that means if , where , then [1], [2] . It is known that for any submodule of , there exists a submodule of , such that , hence is a closed submodule of , is called a closure of [3]. Asgari [4] introduced the notion of t-essential submodule, where a submodule of is called t-essential (denoted by if whenever , implies , where is the second ISSN: 0067-2904
is called -torsion if . Asgari introduced the concept of t-closed submodule where a submodule is called t-closed ( if has no proper tessential extension in [4]. It is clear that every t-closed submodule is closed, but the converse is not true. However, under the class of nonsingular, the two concepts are equivalent. Asgari [5] stated that for any submodule of , there exists a t-closed submodule of such that . is called a t-closure of . A module is called extending if for every submodule of there exists a direct summand ) such that [6]. Equivalently, is an extending module if every closed submodule is a direct summand. As a generalization of extending modules, Asgari [4] introduced the concept of t-extending module, where a module is t-extending if every t-closed submodule is a direct summand. Equivalently, is t-extending if every submodule of is t-essential in a direct summand. The notion of a strongly extending module is introduced in another study [7], which is a subclass of the class of extending module, where an -module is called strongly extending if each submodule of is essential in a fully invariant direct summand of , and a submodule of M is called fully invariant if for each , [8]. A submodule of an -module is called stable if for each -homomorphism [9]. It is clear that every stable submodule is fully invariant but not conversely. An -module is fully stable if every submodule of is stable [9]. An -module is called strongly t-extending if every submodule is tessential in a stable direct summand. Equivalently, is strongly t-extending if every t-closed submodule is a fully invariant direct summand [10]. Saad [7] introduced the stable extending ( Sextending) modules as a generalization of FI-extending modules. An R-module is called stable extending (S-extending) if every stable submodule of is essential in a direct summand of . A ring R is left (right) S-extending if R is S-extending left (right) R-module and is called FI-extending if every fully invariant submodule of is essential in a direct summand of [11] In this paper, we introduce the concepts of t-stable extending and strongly t-stable extending modules. The class of t-stable extending modules contains the class of stable extending, and the class of strongly t-stable contains the class of t-stable extending and it is contained in the class of strongly textending. In section two we study t-stable extending modules and their relationships with other related modules. Among other results in this section, we prove that an -module is a t-stable-extending -module if and only if for each stable submodule of , there is a decomposition such that and . An -module is t-stable extending if and only if for each stable submodule of , there exist such that and where is the injective hull of . Let be a stable injective relative to a stable submodule . If is t-stable extending, then so is . In section three, we study strongly t-stable extending modules. Many properties are given.

T-Stable-extending Modules
In this section we introduce the concept of t-stable extending modules which is a generalization of -extending modules. First we give the following definitions. Definition 2.1: An -module is called t-stable extending if every stable submodule of is tessential in a direct summand. A ring is called right t-stable extending if is a right t-stable extending -module.
Recall that an -module is t-uniform if every submodule of is t-essential in [12]. As a generalization of t-uniform module, we present the following concept. Definition 2.2: An -module is called stable-t-uniform if every stable submodule of is t-essential in .

Remarks and Examples 2.3:
(1) It is clear that every S-extending module (or t-extending module) is t-stable extending, for example: (i)For arbitrary -module , is t-extending [4], so it is t-stable extending. Also as -module is S-extending, so it is t-stable extending.
Recall that an -module is called t-continuous if satisfies the following is t-extending, and every submodule of which contains and isomorphic to direct summand of is itself a direct summand [3]. Hence, every t-continuous module is t-stable extending. Hence, we can give the following examples: (I)By [6, Example 2.6(2)], Let be a -torsion ring (e.g , for a prime number P) and set ( ) . t-continuous T-module. It follows that is a t-stable extending module. However, is not stable extending. Hence not stable extending. (II) Let be a ring and be an -module and . The -module is t-continuous [6, Example 2.6(1)], so it is t-stable extending. In particular if as -module. Then is t-stable (2) Let be a nonsingular -module. Then is -extending if and only if is -stable extending. Proof: since M is non-singular, then the two concepts essential and t-essential coincide [5]. Hence the two concepts, S-extending and t-stable extending, are equivalent.

) Every FI-t-extending is t-stable-extending where
is FI-t-extending if every fully invariant is tessential in a direct summand. Proof: Let be a stable submodule of . Then is fully invariant, hence is t-essential in a direct summand. (5) The converse of (4) holds if is FI-quasi-injective, where an -module is called FI-quasiinjective if for each fully invariant submodule N of M, each R-homomorphism f: N M can be extended to an R-endomorphism g: M M [7]. Proof: Let be a fully invariant submodule of . By [7, Proposition 3.1.19] is stable. Hence by tstable extending property of , is t-essential in direct summand. Thus is a FI-t-extending. (6) -stable extending module need not be extending, for example the -module is not extending but it is S-extending by [7, Remarks and Examples 3.1.3(3)] hence it is t-stable extending. (7) Every stable t-uniform (hence every t-uniform) is t-stable extending. Proof: Let be a stable submodule of . Hence . But , so is t-essential in a direct summand.
Recall that an -module is called an S-indecomposable if (0) , are the only stable direct summand.
is S-extending and S-indecomposable if is S-uniform. An -module is called stable uniform (shortly, -uniform) if every stable submodule of is essential in  [7]. However we have: Proposition 2.4: If is t-stable extending and indecomposable, then is stable t-uniform. Proof: Let be a stable submodule in . Then for some . Since is indecomposable, . Thus and so is a t-stable uniform. Note that a stable t-uniform module does not imply indecomposable, for example as -module is stable t-uniform, but is not indecomposable. Also, is not S-indecomposable. Proposition 2.5: Let be an -module. If is t-stable extending, then every stable t-closed submodule is a direct summand and the converse holds if every t-closure of stable submodule is stable. Proof: Let be a stable t-closed submodule. Since is t-stable extending, for some . Hence , since is a t-closed. Now if is a stable submodule of , then , where is a t-closure of [5,Lemma 2.3]. By hypothesis, is stable, and so is stable t-closed, which implies . Thus is t-essential in a direct summand and is t-stable extending.

Proposition 2.6:
Let be an -module which satisfies that the t-closure of any submodule is stable. Then M is t-stable extending if and only if t-extending. Proof:  Let be a t-closed of . Hence is a t-closure of and so by hypothesis, is stable. But is t-stable extending, so there exists such that . Thus because is tclosed and so is t-extending.  If is t-extending, then by Remarks and Examples 2.3(1), is t-stable extending. Corollary 2.7: Let be a fully stable -module. Then the following statements are equivalent: (1) is a t-stable extending module; (2) is a t-extending module ; is a strongly t-extending module. Proof: Since is a fully stable -module, and the t-closure of any submodule of M is stable . Then (1)  (2) follows by Proposition 2.6.
(1)(3) Let . Since is fully stable, then is stable. Hence is t-essential in a direct summand . But is stable in . Then is t-essential in a stable direct summand and so is strongly t-extending.

Proposition 2.8: Let
be an -module that satisfies that the t-closure of any submodule is stable. Then the following statements are equivalent: (1) is a t-stable extending module; (2) Every stable t-closed submodule of is a direct summand; (3) Every stable submodule is t-essential in stable direct summand. Proof: (1)(2) Let be a stable t-closed submodule. Condition (1) implies is t-essential in a direct summand . Hence since is a t-closed. (2)(3) Let be a stable submodule in . Then has a t-closure ; such that and is a t-closed. But is stable by hypothesis , so that is t-closed stable. Then by condition (2) and hence is t-essential in a stable direct summand.  (2), it is t-stable extending.
By applying Theorem 2.12, each of (for each prime number P) as -module is t-stable extending. Not that and are not extending. Note that by [7, Corollary 3.2.4] every finitely generated -module is S-extending, hence it is t-stable extending. Proposition 2.13: Let be an -module which satisfies that the t-closure of any submodule is stable. If is t-stable extending, then every direct summand is t-stable extending. Proof: Let . Since is t-stable extending, then is t-extending by Proposition 2.6. Hence is t-extending by [4, Proposition 2.14(1)]. It follows that is FI-t-extending and hence by Remarks and Examples 2.3(3), is t-stable extending. Corollary 2.14: Let be a fully stable -module. If is t-stable extending, then every direct summand is t-stable extending.
Recall that an R-module has the summand intersection property (SIP) if the intersection of two direct summands of M is a direct summand [13]. Since S-extending and t-stable extending are equivalent in the class of nonsingular modules, thus we have every direct sumand of t-stable extending module (where is nonsingular with SIP) is t-stable extending module. Also, we have by [2, Corollary 3.2.7, Corollary 3.2.8 and Corollary 3.2.9] the following: 1-Let be a nonsingular SS-module (that is every direct summand is stable). If t-stable extending, then every direct summand is t-stable extending.

2-
Every direct summand right ideal of a nonsingular t-stable extending commutative ring is tstable extending.

3-
Every direct summand of nonsingular cyclic -module is t-stable extending. An R-module M is called stable-injective relative to X (simply, S-X-injective) if for each stable submodule A of X, each R-homomorphism f: A M can be extended to an R-homomorphism g: X M.  [7, Definition 3.2.10].
By using the procedure of the proof of Theorem 3.2.14 [7], we have the following Lemma. Lemma 2.15: Let be a stable injective module relative to a stable submodule of . If such that is a stable in , then is stable in . Proof: Let . Since is stable injective relative to , there exists an -homomorphism such that where is the inclusion mapping from into . It follows that , since is stable in . So ; that is . But is stable in , so that . Thus and is stable in . Proposition 2.16: Let be a stable injective relative to a stable submodule . If t-stable extending, then so is . Proof: To prove is t-stable. Let be a stable submodule of . By Lemma 2.15, A is stable in M. Since M is t-stable extending, there exists such that it follows that for some and so by (5, Corollary 1.3]

Strongly t-stable extending modules
In this section, we extend the notion of t-stable extending modules into strongly t-stable extending modules. We study these classes of modules and their relations with some related concepts. Definition 3.1: An -module is called strongly t-stable extending if each stable submodule of . is t-essential in a stable direct summand.

Remarks and Examples 3.2:
(1) It is clear that every strongly t-stable extending is t-stable extending (2) Every strongly t-extending (hence every -torsion) module is strongly t-stable extending. In particular, each of -module where is a positive integer is strongly t-extending (see [ 10,Example 3.3]. Thus is strongly t-stable extending. (3) The converse of (2) is not true as the following example shows: Let be the -module . Let be a stable submodule of . Then , where is stable in by Lemma 2.11. Since the only stable submodules of are , , then or and hence . Thus is a strongly t-stable extending module. On the other hand, is tclosed(closed) and is not a fully invariant direct summand, since there exists , such that for each and so ( ) .

(4)
Recall that an -module is called weak duo if every direct summand is fully invariant [14]. Let be a week duo. Then strongly t-stable extending if and only if is a t-stable extending module. Proof:  It follows by (1)  Let be a stable submodule of . Then . Since is weak duo, is a fully invariant in and then by [7, Lemma 2.1.6] is stable. Thus is strongly t-stable extending. (5) Let be a fully stable module. Then the following are equivalent: (1) is t-stable extending; is strongly t-stable extending; (4) is strongly t-extending; (6) Every stable t-uniform module is strongly t-stable extending. (7) If is S-indecomposable and is strongly t-stable extending, then is a stable t-uniform. Proof: Let be a stable submodule of . Since is strongly t-stable extending, , is a fully invariant in . Then by[7,Lemma 2.1.6], is stable in , but is S-indecomposable, so . Thus and is a stable t-uniform. (8) If is S-uniform, then is strongly t-stable extending and is S-indecomposable. (9) Let be an indecomposable module. Then is strongly t-stable extending if and only if is tstable extending. (10) If is a FI-t-extending, then is strongly t-stable extending. The converse holds if is FIquasi injective. Proof: Let be a stable submodule of . Then is fully invariant, hence by [11, Theorem 2.2 (1)  (7)] is t-essential in a fully invariant direct summand, say . By [7,Lemma 2.1.6] is stable. Thus is strongly t-stable extending. Proposition 3.3:Let be an -module which satisfies that the t-closure of any submodule is stable. Then the following statements are equivalent: is strongly t-stable extending; is t-stable extending; is t-extending; (4) Every stable t-closed is a direct summand; is strongly t-extending. Proof: (1)  (2) Let N be a stable submodule of N. Then by definition of strongly t-stable extending, N is a t-esential in a fully invariant direct summand. Thus M is t-stable extending.
(3)  (4) Since M is t-extending , every t-closed is a direct summand, so it is clear that every stable t-closed is a direct summand.
(4)  (1) Let be a stable submodule of . Then there exists a t-closure of say such that . By hypothesis, is stable t-closed of , hence . Thus is strongly t-stable extending. (5)  (1) It follows by Remarks and Examples 3.2(2).
(1)  (5) Let be a t-closed of . Hence is a t-closure of and so by hypthesis is stable. Since is strongly t-stable extending, for some stable direct summand . It follows that , since is t-closed. Thus is a stable direct summand and is strongly t-extending. Recall that an module is a multiplication module if for each , there exists an ideal of such that [15]. Proposition 3.4: Let be a multiplication t-extending. Then is strongly t-stable extending. Proof: Let be a stable submodule of . Since is t-stable extending, then there exists such that . But is a multiplication module implies is a fully invariant submodule of and so by [7, Lemma2.1.6], is stable. Thus is t-essential in stable direct summand of . Therefore, is strongly t-stable extending. Corollary 3.5: Every cyclic t-stable extending module over a commutative ring is strongly t-stable extending. Corollary 3.6: Every commutative t-stable extending ring is strongly t-stable extending.
The following is a characterization of strongly t-stable extending modules. Now we ask the following: Is the property of being strongly t-stable extending inherit to a submodule? First we give the following Definition 3.9: An -module is said to be stable-injective if is stable-injective to ( is S-Ninjective), where is any -module. Theorem 3.10: Let be a stable-injective -module. If is strongly t-stable extending, then every stable submodule of is strongly t-stable extending. Proof: Let be a stable submodule of . To prove is strongly t-stable extending, let be a stable submodule of . Since is stable-njective, then stable-injective relative to and hence by Lemma 2.15, stable submodule of . Now is strongly t-stable extending and is stable in imply there exists a stable direct summand such that . Thus for some . Since is stable in , where is stable of , is stable of by Lemma 2.11. Now implies by [3,Corllary 1.3]. But , so that . We claim that is stable in . Since is stable of and is stable in , then is stable of by Lemma 2.15. But is stable in and imply is stable in . Proposition 3.11: Let be an -module which satisfies that the t-closure of any submodule is stable. If is strongly t-stable extending, then every direct summand is strongly t-stable extending. Proof: Let . Since satisfies that the t-closure of any submodule is stable, then by (Proposition 3.3) is strongly t-extending and so by [8,Theorem 3.5] is strongly t-extending. Thus by Remarks and Examples 3.2(2), is strongly t-stable extending. Corollary 3.12: Let be a fully stable -module. If is strongly t-stable extending, then every direct summand is strongly t-stable extending.