STRONGLY T-SEMISIMPLE MODULES AND STRONGLY T-SEMISIMPLE RINGS

In this paper, we introduce the notions of strongly t-semisimple modules and strongly t-semisimple rings as a generalization of semisimple modules, rings respectively. We investigate many characterizations and properties of each of these concepts. An R-module is called strongly t-semisimple if for each submodule N of M there exists a fully invariant direct summand K such that K t-essential in N. Also, the direct sum of strongly t-semisimple modules and homomorvarphic image of strongly t-semisimple is strongly t-semisimple. A ring R is called right strongly t-semisimple if RR is strongly t-semisimple. Various characterizations of right strongly t-semisimple rings are given. AMS Subject Classification: 16D10, 16D70, 16D90, 16P70


Introduction
Through this paper R be a ring with unity and M is a right R-module.Let Z 2 (M) be the second singular (or Goldi torsion) of M which is defined by Z(M/(Z(M ))) = (Z 2 (M ))/(Z(M )) where Z(M ) is the singular submodule of [8].,A submodule A of an R-module M is said to be essential in M (denoted by A ≤ ess M ), if A W (0) for every nonzero submodule W of M .Equivalently A ≤ ess M if whenever A W = 0, then W = 0 [9],Asgari and Haghany [4] introduced the concept of t-essential submodules as generalization of essential submodules.A submodule N of M is said to be t-essential in M (denoted by (N ≤ tes M ) if for every submodule B of M , N B ≤ Z 2 (M ) implies that B ≤ Z 2 (M ).It is clear that every essential submodule is t-essential, but not conversely.However, the two concepts are equivalent under the class of nonsingular modules.A submodule N of M is called fully invariant if f (N ) ≤ N for every R-endomorphism f of M .Clearly 0 and M are fully invariant submodules of M [15] .M is called duo module if every submodule of M is fully invariant.A submodule N of an R-module is called stable if for each homomorphism f:N → M ,f (N ) ≤ N .A module is called fully stable if every submodule of M is stable [1].Asgari and Haghany [3] introduced the notion of t-semisimple modules as a generalization of semisimple modules.A module M is t-semisimple if for every submodule N of M , there exists a direct summand K such that K ≤ tes N .In this paper we introduce the notion of strongly t-semisimple modules as a generalization of t-semisimple modules.An R-module is called strongly t-semisimple if for each submodule N of M there exists a fully invariant direct summand K such that K ≤ tes N .It is clear that the class of strongly t-semisimple modules contains the class of t-semisimple.This paper consists of three sections.In Section 2 we introduce the concept of strongly t-semisimple and giving many characterizations and properties of this class of modules.
Section 3, concerns with strongly t-semisimple rings.Several, characterization of commutative strongly t-semisimple ring.Also we give some characterizations of nonsingular strongly t-semisimple ring.First, we list some known result, which will be needed in our work.
Proposition 1.1.(see [2]) The following statements are equivalent for a submodule A of an R-module. (1) Lemma 1.2.(see [4]) Let A λ be submodule of M λ for all λ in a set Λ.
(1) If Λ is a finite and Lemma 1.3.(see [13]) Let R be a ring and let L ≤ K be submodules of an R-module M such that L is a fully invariant submodule of K and K is a fully invariant submodule of M .Then L is a fully invariant submodule of M .Theorem 1.4.(see [3]) The following statements are equivalent for a module M : (1) M is t-semisimple; (2 where M ′ is a non-singular semisimple module; (4) Every nonsingular submodule of M is a direct summand; (5) Every submodule of M which contains Z 2 (M ) is a direct summand.

Strongly t-Semisimple Modules
Definition 2.1.An R-module is called strongly t-semisimple if for each submodule N of M there exists a fully invariant direct summand K such that K ≤ tes N .

Remarks and Examples.
(1) It is clear that every strongly t-semisimple module is t-semisimple, but the convers is not true as we shall see later. ( and (0) is fully invariant.Hence M is strongly t-semisimple.
(3) Every singular module is strongly t-semisimple.Proof.Let M be a singular R-module.Then Z(M ) = M , it follows that Z 2 (M ) = Z(M ) = M .Thus M is Z 2 torsion, hence M is strongly t-semisimple.Thus, in particular Z n as Z-module is strongly t-semisimple for all n ∈ Z + , n > 1 .
(4) The converse of (3) is not true in general, for example Z 4 as Z 4 -module is not singular, but it is Z 2 -torsion, so it is strongly t-semisimple.
(5) If M is t-semisimple module and weak duo (SS-module).Then M is strongly t-semisimple, where M is a weak duo(or SS-module) if every direct summand of M is fully invariant.
Proof.Let N ≤ M , since M is t-semisimple, there exists K ≤ ⊕ M such that K ≤ tes N .But M is SS-module, so K is stable; hence K is fully invariant direct summand.Thus M is strongly t-semisimple,where M is a weak duo(or SS-module) if every direct summand of M is fully invariant.(6) If M is t-semisimple and duo (or fully stable), then M is strongly tsemisimple.
Hence every t-semisimple multiplication R-module is strongly t-semisimple.
(7) If M is cyclic t-semisimple module over commutative ring R then M is a strongly t-semisimple.
Proof.Since M is cyclic module over commutative ring, then M is a multiplication module.Thus M is duo.Therefor the result follows by part (9).
Theorem 2.3.The following statements are equivalent for an R -module M: (1) M is strongly t-semisimple, (2) M Z 2 (M ) is fully stable semisimple and isomorphic to a stable submodule of M, (3 by Proposition (1.1).On the other hand (N K ′ ) ≤ N and N is nonsingular, so (N K ′ ) is nonsingular submodule, and hence N K is nonsingular, which implies that Z 2 ( N K ) = 0.Thus N K = 0 and hence N = K.Therefore N is a fully invariant direct summand, and hence N is a stable direct summand.

1(4)).We claim that M
′ is nonsingular.To explain our assertion, suppose x ∈ Z(M ′ ), so x ∈ M ′ ≤ M and ann(x) ≤ ess R. Hence ann(x) ≤ tes R and this implies x ∈ Z 2 (M ).Thus x ∈ Z 2 (M ) M ′ = (0), thus x=0 and M ′ is a nonsingular.So that by hypothesis, M ′ is a stable direct summand of M and so that On other hand, ′ is a fully stable module and M ′ is stable in M, so that M (Z 2 (M ) is fully stable semisimple and isomorphic to stable submodule of M.
(2) is projective, we get kerπ which is nonsingular semisimple fully stable module.Then M ′ is nonsingular semisimple fully stable .Also M ′ is stable submodule of M by condition (2).
(3) ⇒ (2) By condition (3),M = Z 2 (M ) ⊕ M ′ , where M ′ , is a nonsingular semisimple fully stable module and M ′ is stable in M. It follows that M ) is semisimple fully stable and isomorphic to stable submodule M ′ of M.
(2) ⇒ (5) It follows directly (since (2) ⇔ (3) ⇒ (5) then(2) ⇒ (5)). ( and so . This implies M Z 2 (M ) is semisimple.By condition (5), M (Z 2 (M )) fully stable and isomorphic to stable submodule of M .But ).It follows that M ′ is fully stable module and M ′ is stable in M. Now we shall give some other properties of strongly t-semisimple.Recall that an R-module M is called quasi-Dedekind if Hom( M N , M ) = 0 for all nonzero submodule N of M.Equivantally, M is quasi-Dedkind if for each f ∈ End(M ),f = 0, then kerf = 0 [10] Proposition 2.4.If M is a quasi-Dedekind module, then M is t-semisimple if and only if M is strongly t-semisimple.
Proof.⇒ since M is quasi-Dedekind, then for each f ∈ EndM f = 0, Kerf = 0, and hence kerf is stable and so that by [14], M is SS-module and so that M is strongly t-semisimple by Remarks and Examples 2.2 (8).
⇐ It is clear.
To prove the next result, we state and prove the following Lemma.
Lemma 2.5.Let N be a submodule of M and K is a direct summand of M such that K ≤ N .If K is fully invariant submodule in M, then K is a fully invariant submodule in N. Proof.To prove K is a fully invariant submodule of N. Let ϕ : N → N be an R-homomorphism, to prove ϕ(K) ≤ K.

Consider the sequence M
Where ρ is the natural projection and i,j are the inclusion mapping.Then (j Proposition 2.6.Every submodule of strongly t-semisimple module is strongly t-semisimple. ).So that K ≤ ⊕ N , and by Lemma (2.5) K is fully invariant submodule of N. Therefore, K is fully invariant direct summand of N such that K ≤ tes W ≤ N .Thus N is a strongly t-semisimple module.Now we consider the direct sum of strongly t-semisimple.First we no-tice that direct sum of strongly t-semisimple module need not be strongly tsemisimple for example: Consider R as R -module R is strongly t-semisimple.But M = R ⊕ R is not strongly t-semisimple by Remarks and Examples 2.2 (12).However, the direct sum of strongly t-semisimple is strongly t-semisimple under certain condition.Before giving our next result, we present the following lemma.
and N 2 submodules of M 1 and M 2 respectively.As M 1 and M 2 are strongly t-semisimple, then there exist Hence M is strongly t-semisimple.Now we shall give other characterizations of strongly t-semisimple module.
Proposition 2.9.The following statements are equivalent for a module M, such that any direct summand has a unique complement: (1) M is strongly t-semisimple, (2) For each submodule N of M, there exists a decomposition M = K ⊕ L such that K ≤ N and L is stable in M and N L ≤ Z 2 L, (3) For each submodule N of M, ( Hence K ≤ tes N and so that M is strongly t-semisimple.Definition 2.10.(see [7]) An R-module M is called comultiplication if ann M ann R N = N for every submodule N of M.   Proof: ⇐It is clear.⇒ Let N≤ M, letϕ ∈ End M .Since M is scalar, there exists r ∈ R such that ϕ(x) = xr,for all x ∈ M .Hence ϕ(N ) = N r ≤ N and so that N is fully invariant submodule.Thus M is duo.But M is duo and t-semisimple implies M is strongly t-semisimple by Remarks and Examples 2.2(6).Proposition 2.17.(2.17):Let M be a duo R-module.Then the following statements are equivalent (1) Every R-module is t-semisimple and Z 2 (M ) is projective.
(2) Every R-module is strongly t-semisimple and Z 2 (M ) is projective.
Proof: (1) ⇒ (3) Let M be an R-module.Then M is t-semisimple by hypothesis.Hence M = Z 2 (M ) ⊕ M ′ , where M ′ is a nonsingular semisimple.It follows that M ′ is projective, but by hypothesis Z 2 (M ) is projective.Thus M is projective, that is every R-module is projective and so by [11, is projective (by [10, Coroallary 1.25,P.35] .Now let π : M → M/(Z ( 2)(M )) be the natural epiomorphism and as M (Z 2 (M ))

Lemma 2 . 11 .
Every comultiplication module is fully stable.Proof.Let M be a comultiplication R-module.Then ann M ann R N = N for all N ≤ M .Hence ann M ann R (xR) = xR for all cyclic submodules xR in M .Thus M is fully stable, [2, Corollary(3.5)].

Corollary
2.12.Let M be a comultiplication R-module.Then M is t-semisimple if and only if M is strongly t-semisimple.Proof.⇐ It is clear.⇒ It follows directly by Lemma (2.11) and Remarks and Examples 2.2(6).Recall that an R-module M is called a principally injective if for any a ∈ R, any homomorphism f : Ra → M extends to an R-homomorphism from R R to M [12].Corollary 2.13.Let M be a principally injective.Then M is t-semisimple if and only if M strongly t-semisimple.Proof.⇐ It is clear.⇒ M is principally injective implies that ann M ann R (x) = (x) for each x ∈ R. Hence by [2, Corollary(3.5)]M is fully stable.Then by Remark and Examples 2.2(5), M is strongly t-semisimple.Corollary 2.14.(2.14): M is injective R-module.Then M is t-semisimple R-module if and only if M is strongly t-semisimple.Definition 2.15.(2.15) [12]: An R-module is called scalar if for all ϕ ∈ End M , there exists r ∈ R such that ϕ(x) = xr for all x ∈ M , where R is a commutative ring.Proposition 2.16.(2.16):Let M be a scalar R-module.Then M is t-semisimple if and only if M is strongly t-semisimple, where R is commutative.