On left weakly jointly prime (R,S)-modules

Let R and S be commutative rings and M an (R,S)-module. A proper (R,S)-submodule P of M is called left weakly jointly prime if for each (R,S)-submodule N of M and elements a,b of R such that abNS\subseteq P implies either aNS\subseteq P or bNS\subseteq P. This paper defines left weakly jointly prime (R,S)-modules and presents some of their properties. On the other hand, a ring R is called fully prime if each proper ideal of R is prime. We extend this fact to (R,S)-modules. An (R,S)-module M is called fully left weakly jointly prime if each proper (R,S)-submodule of M is left weakly jointly prime. Moreover, we present some properties of fully left weakly jointly prime (R,S)-modules. At the end of this paper, we present our main results about the necessary and sufficient conditions for an arbitrary (R,S)-module to be fully left weakly jointly prime.

کلیدواژه ها:

Fully prime ، Prime module ، (R ، S)-submodule ، Weakly prime