(m,n)-Closed submodules of modules over commutative rings

Let [Formula: see text] be a commutative ring and [Formula: see text], [Formula: see text] be positive integers. We define a proper submodule [Formula: see text] of an [Formula: see text]-module [Formula: see text] to be [Formula: see text]-closed if for [Formula: see text] and [Formula: see text], [Formula: see text] implies [Formula: see text] or [Formula: see text]. This class of submodules lies properly between the classes of prime and primary submodules. Many characterizations, properties and supporting examples concerning this class of submodules are provided. The notion of [Formula: see text] -modules is introduced and characterized. Furthermore, the [Formula: see text]-closed avoidance theorem is proved. Finally, the [Formula: see text]-closed submodules in amalgamated modules are studied.