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.