Stable Semisimple Modules, Stable t- Semisimple Modules and Strongly Stable t-Semisimple Modules

Throughout this paper, three concepts are introduced namely stable semisimple modules, stable tsemisimple modules and strongly stable t-semisimple. Many features co-related with these concepts are presented. Also many connections between these concepts are given. Moreover several relationships between these classes of modules and other co-related classes and other related concepts are introduced.

Hadi I-M.A. and Shyaa F.D. in (3) extend the notion of t-semisimple in to strongly tsemisimple modules and studied them.
In (4), they introduced and studied these concept FI-semisimple modules, where "an −module is called FI-semisimple if every fully invariant submodule is a direct summand" (4)." is called FI-t-semisimple module if for each fully invariant submodule of , there exists ≤ ⊕ such that ≤ " (4). " is called strongly FI-tsemisimple if for each fully invariant submodule of , there exists a fully invariant submodule of with ≤ " (4). "A submodule of is called fully invariant if for each endomorphism (i.e. ∈ ( )), ( ) ⊆ " (1). " is called stable if for each homomorphism : → , ( ) ⊆ (5). " is called Duo (fully stable) if every submodule is fully invariant (stable)" (6) and (5). Obviously "every stable submodule is fully invariant but the converse is not true in general", see (5), (7). This motivate us to introduce and study these types of modules: stable semisimple, stable t-semisimple and strongly stable t-semisimple modules. Section 2 is devoted for studying stable semisimple modules. The direct sum of stable semisimple modules is stable semisimple (see proposition 3). However a direct summand of stable semisimple inherits the property under certain condition (see proposition 4). Also, stable submodules inherit the property if the module is stable injective (see proposition 5).
In Section 3, the stable t-semisimple modules are introduced and studied which as a generalization of t-semisimple modules and also a generalization of FI-t-semisimple modules. The direct sum of stable t-semisimple modules 1 and 2 is stable t-semisimple and the converse hold if = 1 ⊕ 2 is stable injective and 1 + 2 = (see Theorm 1). Beside this, many characterizations of stable t-semisimple module (with certain conditions) are presented.
In Section 4, strongly stable t-semisimple is introduced and studied. This concept is a generalization of strongly t-semisimple, also a generalization of strongly FI-t-semisimple. Many connections between this concept and other concepts such as stable semisimple, 2 −torsion are given. Strongly stable t-semisimple modules and strongly FI-t-semisimple modules are coincide under certain conditions (see Remarks and Examples 3(6), (7)). The direct sum of two strongly stable t-semisimple modules 1 , 2 with 1 + 2 = is strongly stable t-semisimple, and the converse hold if = 1 ⊕ 2 is stable injective. (Theorem 3). Also every stable direct summand of strongly stable t-semisimple module is strongly stable-t-semisimple if is stable-injective (see Proposition 4). Many other results are given in section 4.

Stable Semisimple:
In this section, the stable semisimple modules are introduced and studied.

Definition 1: An R-module
is called stable semisimple (briefly s-semisimple) if every stable submodule of is a direct summand. A ring is s-semisimple if every stable ideal of is a direct summand of .
Note that an R-module is s-semisimple module if for each stable submodule of , there exists ≤ ⊕ such that ≤ .

Remarks and Examples 1:
1. Every semisimple module is s-semisimple, but the converse may be not correctly, for instance the − module is s-semisimple since it has only two stable submodules namely (0), and they are direct summands, and Z is not semisimple. 2. Since every stable submodule is fully invariant, then every FI-semisimple module is ssemisimple. However s-semisimple module may be not FI-semisimple; as: as − module is s-semisimple and it is not FI-semisimple since every proper non zero submodule of is fully invariant but it is not direct summand. 3. "An R-module is called stable extending ( −extending) if every stable submodule of is essential in a direct summand " (7).
Proposition 1: Let be a s-semisimple and is a stable submodule of . Then is s-semisimple.
Proof: Let be a stable submodule of where U≤M and U contains N . By Lemma 1, is a stable submodule of . But is s-semisimple, hence ≤ ⊕ ; that is ⊕ V = M for some ≤ . This implies = ⊕ + and so that ≤ ⊕ and is s-semisimple.
Corollary 1: Let : → ′ be an epimorphism such that is a stable submodule of . If is s-semisimple, then ′ is s-semisimple.
The following proposition shows that the property of s-semisimple inhirts to direct summands, under certain conditions. First the following Lemma is given.

Proposition 5:
Let be a s-injective −module. If is s-semisimple module, then every stable submodule of is s-semisimple. Proof: Let be a stable submodule of and let be a stable submodule of , then by (8,Lemma: 2.15).

Stable t-Semisimple Modules:
In this section, the concept of stable tsemisimple modules are introduced and studied, which is a generalization of s-semisimple modules. Also it is a generalization of t-semisimple modules and FI-t-semisimple modules.

Remarks and Examples 2:
1. clearly every s-semisimple module is s-tsemisimple, but the converse is not true in general, for example: the Z-module 4 is s-tsemisimple, since for each ≤ 4 , is stable and (0 ≤ because 0 + 2 ( ) = ≤ ), see (2, proposition1.1). 2. Every Singular (and hence 2 -torsion) module is s-t-semisimple, since for each ≤ , (0) + 2 ( ) = (0) + = ≤ and hence (0) ≤ by (2, proposition.1.1). 3. Every t-semisimple module is s-t-semisimple, but the converse may be not true, for example: as −module is not t-semisimple (since But is s-t-semisimple since It is ssemisimple. Also = ⊕ 2 as −module is s-semisimple by part 2., So it is s-t-semisimple, but is not t-semisimple since Note that under the class of fully stable modules the two notions (t-semisimple) and (st-semisimple)module are equivalent. Also they are equivalent under the class of comultiplication modules, since "every comultiplication modules is fully stable", see (9,lemma,1.2.12,p.39). 4. Every FI-t-semisimple is s-t-semisimple, but the convers may be false, as the following example shows: as −module is s-t-semisimple and it is not FI-t-semisimple by (

Proposition 6:
Let be an s-injective module. If is a s-t-semisimple module, then every stable submodule of is s-t-semisimple. Proof: Let be a stable submodule of and let be a stable submodule of . Since is stable injective, is stable in by (8,Lemma: 2.15). It follows that there exists ≤ ⊕ and ≤ , since is s-t-semisimple. Hence = ⊕ T for some ≤ and so that = ( ⊕ ) ∩ = ⊕ (T ∩ U), thus ≤ ⊕ and hence U is a stable tsemisimple.
Recall that for any submodule of , is contained in a t-closed submodule of , such that ≤ by (10,Lemma 2.3). is called a tclosure of (10).

Proposition 8: Let
be an s-injective module such that a complement of 2 ( ) is stable and a tclosure of stable submodule is stable. If is s-tsemisimple, then is t-stable extending. Proof: By Theorem 2 ((1)⇒(5)), each stable submodule of with 2 ( ) ⊆ , ≤ ⊕ . Hence every t-closed stable submodule is direct summand, since every t-closed submodule contains 2 ( ). On the other hand, by hypothesis a t-closure of stable submodule is stable, hence by (8, proposition. 2.5), is t-stable extending.

Proposition 9:
Let be a s-injective module such that a complement of stable submodule is stable and a t-closure of stable submodule is stable. If is st-semisimple, then is s-t-semisimple for each stable t-closed submodule C. Proof: By Proposition 8, is t-stable extending, so by (8, Proposition 2.5), every stable t-closed submodule is a direct summand of . Hence = ⊕ for some ≤ . It follows that is a complement of and hence is a stable submodule of . Thus by Proposition 6, is s-t-semisimple. But ≅ , so is stable t-semisimple.

Strongly Stable t-semisimple Modules:
Our concern in this section is extending the notions of s-t-semisimple modules into strongly stable t-semisimple. Also this concept is a generalization of the concept strongly t-semisimple which is introduced in (3) where " an -module M is strongly t-semisimple if for each submodule of ,there exists a fully invariant direct summand (hence stable direct summand) of such that ≤ " (3).

Definition 3:
An -module is called strongly stable t-semisimple ( shortly s-s-t-semisimple) if for each stable submodule of , there exists a stable direct summand of with ≤ .

Remarks and Examples 3:
1) Every s-semisimple module is s-s-tsemisimple but not conversely as can see by the example 12 as -module is s-s t-semisimple, but not stable semisimple.
2) Every strongly t-semisimple is s-s-t-semisimple, but the converse may be not achieved , for example: Let = ⊕ as -module. Since has only two stable submodules which are and (0), so is s-semisimple and hence by (1) is st-semisimple. However is not strongly tsemisimple since 2 ( ) ≃ is not t-semisimple (9,Ex.4,p.26).
3) Every 2 -torsion module is s-t-semisimple by (3,Rem &Ex. (3)), so It is s-s t-semisimple. Note that 4 as 4 -module is s-t-semisimple but not 2 -torsion. 4) Every s-s-t-semisimple implies s-t-semisimple. 5) "An -module is called strongly FI-tsemisimple if for each fully invariant submodule of , there exists a fully invariant direct summand of , with ≤ " (4), Then every strongly FI-t-semisimple is s-s tsemisimple, but the converse is not achieved for example: the -module is s-s t-semisimple, but is not strongly FI-t-semisimple since if = , ∈ , > 1 . then (0) is the only direct summand of such that (0) ⊆ but (0) ≰ . 6) Let be a FI-quasi-injective -module. Then is s-s -t-semisimple if and only if strongly FI-tsemisimple. 7) Let be a fully stable -module. Then the following statements are equivalent. (1) is strongly t-semisimple.
is strongly FI-t-semisimple.
is t-semisimple. Proof: it is clear  ≤ . Now, since K ≤ ⊕ M then ⊕ ′ = and so = ⊕ ( ′ ∩ ), thus K ≤ ⊕ M. but by (9,Rem 1.1.36), K is stable in N. Thus K is a stable direct summand of N with ≤ , so that is s-s-t-semisimple.

Corollary 2:
Let be an s-injective. If is s-s-tsemisimple, then every nonsingular stable submodule of is s-s-t-semisimple. Proof: Let be a nonsingular stable submodule of . Since is s-s-t-semisimple, then is stable tsemisimple by Rem &Ex 3(4). And by Theorem 3.5(1⇒4), N ≤ ⊕ M. thus N is s-s-t-semisimple by Proposition 4.

Corollary 3:
For an s-injective -module which satisfies (a Complement of 2 ( ) is stable). If is s-s-t-semisimple, then every stable submodule of which contains 2 ( ) is s-s-t-semisimple. Proof: since is s-s-t-semisimple module, then by Theorem 3.5(1⇒5), N ≤ ⊕ M. it follows that is ss-t-semisimple by Proposition 4.