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; / : LðMÞ ! LðMÞ[ ;f g be a reduction function and d : LðRÞ ! LðRÞ be an expansion function, where LðMÞ is the lattice of all submodules of M and LðRÞ is the lattice of all ideals of R: A proper submodule N of M is said to be a /-d-primary submodule of M if whenever am 2 N /ðNÞ for some a 2 R and m 2 M; then either a 2 dððN : MÞÞ or m 2 N: Many properties, characterizations and examples of /-d-primary submodules are given. Keywords Prime submodule Primary submodule /-Prime /-d-Primary submodule