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 ⊆ P implies either aNS ⊆ P or bNS ⊆ 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