On weakly classical 1-absorbing prime submodules

Abstract

In this paper, we study weakly classical 1-absorbing prime submodules of a nonzero unital module M over a commutative ring R having a nonzero identity. A proper submodule N of M is said to be a weakly classical 1-absorbing prime submodule, if for each m is an element of M and nonunits a, b, c is an element of R, 0 not equal abcm is an element of N implies that abm is an element of N or cm is an element of N. We give various examples and properties of weakly classical 1-absorbing prime submodules. Also, we investiage the weakly classical 1-absorbing prime submodules of tensor product F circle times M of a (faithfully) flat R-module F and any R-module M. Also, we prove that if every proper submodule of an R-module M is weakly classical 1-absorbing prime, then Jac(R)3 M = 0. In terms of this result, we characterize modules over local rings in which every proper submodule is weakly classical 1-absorbing prime.