A variation of supplement submodules

In this paper, we introduce and study [Formula: see text]-supplement submodules and [Formula: see text]-almost injective modules. A submodule [Formula: see text] of a module [Formula: see text] is called a [Formula: see text]-supplement in [Formula: see text] if there exists a submodule [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text] is small and fully invariant in [Formula: see text]. A module [Formula: see text] is called [Formula: see text]-almost injective, if [Formula: see text] is a [Formula: see text]-supplement submodule of every module which contains [Formula: see text]. We investigate several properties of these notions and present new characterizations of some classes of rings in terms of [Formula: see text]-supplement submodules.