On φ-δ-Primary Submodules

Abstract

In this paper, we introduce and study a new class of submodules which unify the concepts of prime and primary submodules. Let M be a unital module over a commutative ring R,ϕ:L(M)→L(M)∪{∅} be a reduction function and δ: L(R) → L(R) be an expansion function, where L(M) is the lattice of all submodules of M and L(R) is the lattice of all ideals of R. A proper submodule N of M is said to be a ϕ-δ-primary submodule of M if whenever am∈ N- ϕ(N) for some a∈ R and m∈ M, then either a∈ δ((N: M)) or m∈ N. Many properties, characterizations and examples of ϕ-δ-primary submodules are given.