In this paper, we introduce the concept of a quasi-radical semi prime submodule. Throughout this work, we assume that is a commutative ring with identity and is a left unitary R- module. A proper submodule of is called a quasi-radical semi prime submodule (for short Q-rad-semiprime), if for , ,and then . Where is the intersection of all prime submodules of .
Let R be a commutative ring with non-zero identity element. For two fixed positive integers m and n. A right R-module M is called fully (m,n) -stable relative to ideal A of , if for each n-generated submodule of Mm and R-homomorphism . In this paper we give some characterization theorems and properties of fully (m,n) -stable modules relative to an ideal A of . which generalize the results of fully stable modules relative to an ideal A of R.
In this paper we introduce the notions of bi-ideal with respect to an element r
denoted by (r-bi- ideal ) of a near ring , and the notion fuzzy bi- ideal with respect
to an element of a near ring and the relation between F-r-bi-ideal and r-bi-ideal of
the near ring, we studied the image and inverse image of r-bi- ideal under
epimomorphism ,the intersection of r-bi- ideals and the relation between this ideal
and the quasi ideal of a near ring, also we studied the notion intuitionistic fuzzy biideal
with respect to an element r of the near ring N, and give some theorem about
this ideal .
The main objective of this research is to study and to introduce a concept of strong fully stable Banach -algebra modules related to an ideal.. Some properties and characterizations of full stability are studied.