Nil Versions of Prime Submodules

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be an [Formula: see text]-module. A proper submodule [Formula: see text] of [Formula: see text] is said to be prime if for any [Formula: see text] and [Formula: see text] with [Formula: see text], we have [Formula: see text] or [Formula: see text] ([Formula: see text]). The aim of this paper is to introduce and investigate the notions of lopsided-nil-prime and lopsided-[Formula: see text]-prime submodules of [Formula: see text] as two generalizations of prime submodules of [Formula: see text].