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.
