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 a unital module over noncommutative ring with nonzero identity. This is new class between (weakly submodules) 2-absorbing submodules). Let R be identity $$1\ne 0$$ 1 ≠ 0 M an -module. A proper submdule P said submodule submodule) if for all nonunits $$x,y\in R$$ x , y ∈ R $$m\in M$$ m M xRyRm $$\subseteq $$ ⊆ $$(\left\{ 0\right\} \ne ( P)$$ P ) , then $$xy\in (M:_{R}P)$$ : or P$$ . Various properties characterizations these classes are considered.