Approximaitly Semi-Prime Submodules and Some Related Concepts
Prime submodules
Semi-prime submodules
Quasi-prime submodules
Approximaitly prime submodules
Approximaitly semi-prime submodules
Multiplication module
Socle of modules.
We introduce in this paper the concept of approximaitly semi-prime submodules of unitary left -module over a commutative ring with identity as a generalization of a prime submodules and semi-prime submodules, also generalization of quasi-prime submodules and approximaitly prime submodules. Various basic properties of an approximaitly semi-prime submodules are discussed, where a proper submodule of an -module is called an approximaitly semi-prime submodule of , if whenever , where , and , implies that . Furthermore the behaviors of approximaitly semi-prime submodule in some classes of modules are studied. On the other hand several characterizations of this concept are introduced.
