On ?-1-absorbing primary submodules

Abstract

In this article, we introduce ϕ-1-absorbing primary submodules of modules over commutative rings. Let R be a commutative ring with a nonzero identity and M be a nonzero unital module. ϕ: S(M) → S(M) ∪ {∅} be a function where S(M) is the set of all submodules of M. A proper submodule N of M is said to be a ϕ-1-absorbing primary submodule if whenever abm ∈ N − ϕ(N) for some nonunit elements a, b ∈ R and m ∈ M, then ab ∈ (N:R M) or m ∈ M-rad(N), where M-rad(N) is the prime radical of N. Many properties and characterizations of ϕ-1-absorbing primary submodules are given. We also give the relations between ϕ-1-absorbing primary submodules and other classical submodules such as ϕ-prime, ϕ-primary, ϕ-2-absorbing primary submodules. © 2024, Sidi Mohamed Ben Abdellah University. All rights reserved.