1-absorbing and weakly 1-absorbing prime submodules of a module over a noncommutative ring

Abstract

In this study, we aim to introduce the concepts of 1-absorbing prime submodules and weakly 1-absorbing prime submodules of a unital module over a noncommutative ring with nonzero identity. This is a new class of submodules between prime submodules (weakly prime submodules) and 2-absorbing submodules (weakly 2-absorbing submodules). Let R be a noncommutative ring with a nonzero identity 1ne 0 and M an R-module. A proper submdule P of M is said to be a 1-absorbing prime submodule (weakly 1-absorbing prime submodule) if for all nonunits x,yin R and min M with xRyRmsubseteq P(left{ 0right} ne xRyRmsubseteq P), then xyin (M:_{R}P) or min P. Various properties and characterizations of these classes of submodules are considered.