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

Abstract

AbstractIn 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 $$1\ne 0$$ 1 ≠ 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 $$x,y\in R$$ x , y ∈ R and $$m\in M$$ m ∈ M with xRyRm$$\subseteq $$ ⊆ P$$(\left\{ 0\right\} \ne $$ ( 0 ≠ xRyRm$$\subseteq P)$$ ⊆ P ) , then $$xy\in (M:_{R}P)$$ x y ∈ ( M : R P ) or $$m\in P$$ m ∈ P . Various properties and characterizations of these classes of submodules are considered.