Semisimple Modules Relative to A Semiradical Property

In this paper, we introduce the concept of s.p-semisimple module. Let S be a semiradical property, we say that a module M is s.p - semisimple if for every submodule N of M, there exists a direct summand K of M such that K ≤ N and N / K has S. we prove that a module M is s


Introduction
Throughout this paper, all rings are associative with identity and all modules are unitary left R-modules. Let A be a submodule of a module M. A is called an essential submodule of M (denoted by A ≤ e M) if A ∩ B ≠ 0, ∀ 0 ≠ B ≤ M. A submodule B of M is called a closed submodule of M if B has no proper essential extension. A module M is called an extending module if every submodule of M is essential in a direct summand. Equivalently, every closed submodule of M is a direct summand, see [1], [2], [3]. Let M be a module. Recall that the socle of M (denoted by Soc(M)) is the sum of all simple submodules of M, a module M is called a semisimple if Soc(M) = M. Equivalently a module M is semisimple if and only if every submodule is a direct summand of M, see [1], [4]. Recall that the Jacobson radical of M (denoted by J(M)) is the intersection of all maximal submodules of M. If M has no maximal submodule, we write J(M) = M, see [5].
Let x ∈ M. Recall that ann (x) = {r ∈ R: rx = 0}. For a module M, the singular submodule is defined as follows Z(M) = {x ∈ M | ann x ≤ e R} or equivalently, Ix = 0 for some essential left ideal I of R. If Z(M) = M, then M is called a singular module. If Z(M) = 0, then M is called a nonsingular module. The second singular (or Goldie torsion) submodule of a module M (denoted by Z 2 (M)) is defined by Z(M / Z(M)) = Z 2 (M) / Z(M), see [1], [6].
A submodule A of a module M is called t-essential submodule (denoted by A ≤ tes M) if for any submodule B of M, A ⋂ B ≤ Z 2 (M) implies B ≤ Z 2 (M). A module M is called t-semisimple if for every submodule N of M there exists a direct summand K of M such that K ≤ tes N, see [5]. [7].
A property S is called a radical property if: 1-for every module M, there exists a submodule (denoted by S(M)) such that a-S(M) has S. b-A ≤ S(M), for every submodule A of M such that A has S. 2-If f: M → N is an epimorphism and M has S, then N has S. 3-S(M / S(M)) = 0 for every R-module M, see [8].
A property S is called a semiradical property if it satisfies conditions 1 and 2, see [8].
It's known that each of the following two properties is a radical property, see [8]. While each of the following two properties is a semiradical property (but it is not radical property), see [8] In this paper, S is a semiradical property, unless otherwise stated.

2-s.p -semisimple modules
In this section, we introduce the concept of s.p-semisimple modules and give the basic properties of this module. Also, we illustrate it with some examples.
Definition2.1. Let S be a semiradical property. We say that a module M is s.p -smisimple module if for each submodule N of M, there exists a direct summand K of M such that K ≤ N and N / K has S.

Remarks and Examples2.2.
1-Every semisimple module is s.p -semisimple. The converse is not true in general. Proof. Let N be a submodule of a semisimple module M, then N is a direct summand of M, by [4]. Let K = N, hence S(N / K) = S(N / N) = S(0) = 0 ≅ N / K. Thus M is s.p -semisimple. For example Z 6 as Z 6 -module is s.p -semisimple module.
Recall that a semiradical property S is called hereditary if S is closed under submodules, see [8].
Since M i is s.p -semisimple, then there exists K i is a direct summand of M i such that K i is a submodule of N i and N i / K i has S ∀ i ∈ I. Hence ((⊕ i∈ I N i ) / (⊕ i∈ I K i )) ≅ ⊕ i∈ I (N i / K i ) has S, by [10]. Thus M = ⊕ i∈ I M i is s.p -semisimple.  [8]. But by [15,lem.3.2] Let M be an R-module. M is said to have the summand intersection property (briefly SIP) if the intersection of any two direct summands of M is a direct summand of M, see [16]. Let : R → Rm be a map defined by (r) = rm, for each r ∈ R.It is easy to see that is an epimorphism and Ker ( ) = ann (m). By the first isomorphism theorem, R / ann(m) ≅ Rm. Since M is torsion free module, then ann(m)= 0. Thus R ≅ Rm. But R is indecomposable. Therefore, Rm is indecomposable. Implies that either Rm = K or Rm = Rm ∩ H. Thus either Rm is a direct summand of M or Rm has S.

Proposition2.16. Let R be an indecomposable ring and M be a projective module. If M is s.p -semisimple module, then for every m ∊ M, either Rm is a direct summand of M or Rm has S.
Proof. Assume that M is a projective and s.p -semisimple module and let m ∊ M. Then there exists a direct summand K of M such that K ≤ Rm and Rm / K has S. Let M = K ⊕ H for some submodule H of M, then Rm = K ⊕ (H ∩ Rm), by modular law. But Rm / K ≌ H ∩ Rm, by the second isomorphism theorem. Therefore, H ∩ Rm has S. Now, let : R → Rm be a map defined by (r) = rm, for all r ∊ R. It is clear that is an epimorphism map. Let P: Rm → K be the projection map. Clearly, P : R → K is an epimorphism. Since M is projective, then K is projective by [4]. Therefore, Ker (P ) is a direct summand of R. Since R is indecomposable, then either Ker P = 0 or Ker P = R. Ker (P ) = -1 (Rm ∩ H) = -1 (Rm ∩ H). So either Rm ∩ H = 0 or Rm ∩ H = R. Thus Rm = K or Rm ∩ H = Rm has S.

3-Characterization of s.p -semisimple Modules
In this section, we give various characterizations of s.p -semisimple modules.
We start with the following theorem.  Let M be a module. Recall that M is called an S-generalized supplemented module (or briefly S-GS module), if every submodule of M has S-generalized supplement in M, where S is semiradical property on modules, see [17].

Proposition3.4. Every s.p -semisimple module M is S-GS supplemented module.
Proof. Let M is s.p -semisimple module and N be a submodule of M, then there exists a direct summand K of M such that K ≤ N and N / K has S. Hence, M = K ⊕ K 1 , for some submodule K 1 of M. But K ≤ N, therefore M = N + K 1 . So by modular law, N = K⊕(N ∩ K 1 ), then by the second isomorphism theorem, N / K ≅ N ∩ K 1 has S. Thus N ∩ K 1 ≤ S(K) by [8]. . But B has S. Therefore, K ∩ A 1 has S, by [8]. Hence, K∩A 1 ≤ S(A 1 ). Thus A 1 is an S-generalized supplement submodule of K in M and A 1 is contained in N.
Proposition3.6. Let S be a hereditary property and M be a module. Then the following statements are equivalent 1-M is s.p -semisimple module. 2-Every submodule N of M has S-generalized supplement K in M such that N ∩ K is a direct summand of N.
Proof. 1⇒2) Let N be a submodule of M. Then by the same argument of proof of Proposition 3.4. N has an S-generalized supplement. 2⇒1) Let N be a submodule of M. Then by our assumption N has an S-generalized supplement K in M such that N∩K is a direct summand of N. Hence M = N + K and N ∩ K ≤ S(K). Let N= (N ∩ K) ⊕ L, for some submodule L of N. Then M = (N∩K) + L + K = L + K. But, L ∩ K = N ∩ K ∩ L = 0. Therefore, M = L ⊕ K. By the second isomorphism theorem, N / L ≅ N ∩ K. Since N ∩ K ≤ S(K) and S is hereditary property, then N ∩ K has S by [8] and hence N / L has S. Thus M is s.p-semisimple. Let M be an R-module. Recall that M is called π-projective (or co-continuous) if for every two submodules U, V of M with U + V = M there exists ∈ End(M) with Im ( ) ≤ U and Im (1− f) ≤ V, see [18].

Conclusion
In this work, the concept of s.p-semisimple module is introduced and studied. We also conclude the following: