On φ-δ-Primary Submodules

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,phi:L(M) -> L(M)boolean OR{empty set} be a reduction function and delta: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 phi-delta-primary submodule of M if whenever am is an element of N-phi(N) for some a is an element of R and m is an element of M, then either a is an element of delta((N:M)) or m is an element of N. Many properties, characterizations and examples of phi-delta-primary submodules are given.