primary submodules over a multiplicatively closed subset of a commutative ring

In this paper, we introduce the concept of primary submodules over s which is a generalization of the concept of s-prime submodules. suppose s is a multiplicatively closed subset of a commutative ring r and let m be a unital r-module. a proper submodule q of m with (q :r m) ∩ s = ∅ is called primary over s if there is an s ∈ s such that, for all a ∈ r, m ∈ m, am ∈ q implies that sm ∈ q or san ∈ (q :r m), for some positive integer n. we get some new results on primary submodules over s. furtheremore, we compare the concept of primary submodules over s with primary ones. in particular, we show that a submodule q is primary over s if and only if (q :m s) is primary, for some s ∈ s.