-Semiprime Submodules

Let be a commutative ring with identity and a fixed ideal of and be an unitary -module.We say that a proper submodule of is -semi prime submodule if with . In this paper, we investigate some properties of this class of submodules. Also, some characterizations of -semiprime submodules will be given, and we show that under some assumptions -semiprime submodules and semiprime submodules are coincided.


Introduction
Throughout, represents an associative ring with nonzero identity and a fixed ideal of R and be a unitary -module. The concept of semiprime submodules was introduced and studied in [1980], where a proper submodule of is called a semiprime submodule if for each , k with implies that r [1].Then, many generalizations of semiprime submodules were studied such as weakly semiprime submodules in [2], S-semiprime submodules in [3]and nearly semiprime submodules in [4].
In this paper, we extend the concept of semiprime submodules. Let a fixed ideal of A proper submodule of is called -semiprime if whenever with x implies that We generalize some basic properties of prime and semiprime to -semiprime submodules and give some characterizations of -semiprime submodules.  [9]. By using these concepts we can give the following proposition.

Proposition (1.7):Let a proper submodule of an -module
If is -prime then issemiprime. Proof: Let is -prime submodule of an R-module M, Assume that where Since and is -prime submodule of , then either or In any case, we have . Therefore is -semi prime submodule of . Proposition(1.8):Let a proper submodule of an -module such that is radical ideal If is -primary submodule in , then is an -prime (and hence -semi prime) submodule of . Proof:Let is -primary submodule and is radical ideal . Assume that where suppose m Since is -primary submodule of and m , then √ But is radical, so .Therefore is -prime (and hence -semi prime)submodule of . From proposition (1.8) we get the following: Corollary (1.9):Let a proper submodule of an -module such that is semi prime ideal of If is -primary submodule in , then is an -prime (and hence -semiprime) submodule of . .] The quotient and localization of prime submodules are again prime submodules. But in case of Isemiprime submodules. We give a condition under which the quotient and localization becomes true as we see in the following theorem. Theorem(1.14): Let be an -module. Let be an -semi prime submodule of . Then: 1) Suppose that is a multiplicatively closed subset of such that and Then is an -semiprime submodule of an -module . where is an -semiprime submodule of and = . 2.
where is an -semiprime submodule of and = . Proof.

Suppose that
is an -semiprime submodule of and = Let ( , ) such that ( , ) ( -- . Then and is -semi prime submodule of so a . Therefore (a, b) . So is an I-semiprime submodule of 2. The proof is similar to part (1).

Remark (1.22):Let
Let (i=1,2) with . Let be ideals of respectively with Then all the following types are Isemiprime submodule of 1-where is a proper submodule of with for 2-where is a prime submodule of .

3-
where is a prime submodule of . Proof. 1. Since = and = . Then = ( = I ( = . So -I ( . Thus there is nothing to prove. 2. Let be a prime submodule of . Then is a prime submodule of and hence I-prime (I-semiprime) submodule of by (1.6). 3. The proof is similar to the part (2).