Study on prime and semi-prime rings and modules

Abstract

Let R be a ring. Then a proper ideal P in a ring R is called a prime ideal of R if for any ideals I, J of R with IJ ⊂ P, then either I ⊂ P or J ⊂ P. A ring R is called a prime ring if 0 is a prime ideal. Let M be a right R-module and S End (M), R = its endomorphism ring. A submodule X of M is called a fully invariant submodule of M if for any f ∈ S, we have f (X ) ⊂ X. Let M be a right R-module and P, a fully invariant proper submodule of M. Then P is called a prime submodule of M if for any ideal I of S , and any fully invariant submodule X of M, I (X ) ⊂ P implies I (M) ⊂ P or X ⊂ P. A fully invariant submodule X of a right R-module M is called a semi-prime submodule if it is an intersection of prime submodules. An ideal P in a ring R called a semi-prime ideal if it is an intersection of prime ideals. A ring R is called a semi-prime ring if 0 is a semi-prime ideal. This study describes some properties of prime and semi-prime ideals in associative rings modifying the results on prime and semi-prime Goldie modules investigated in [15]. The structures of prime and semi-prime rings are also available in this study. Finally, some properties of prime and semi-prime submodules as a generalization of prime and semi-prime ideals in associated rings are also investigated.