Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On weakly classical 1-absorbing prime submodules

Zeynep Yılmaz Uçar, Bayram Ali Ersoy, Ünsal Tekir, Suat Koç, Serkan Onar

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 $m\in M$ and nonunits $a,b,c\in R,$ $0\neq abcm\in N$ implies that $abm\in N$ or $cm\in N$. 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 $F\otimes M$ 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.

On weakly classical 1-absorbing prime submodules

Abstract

In this paper, we study weakly classical 1-absorbing prime submodules of a nonzero unital module over a commutative ring having a nonzero identity. A proper submodule of is said to be a weakly classical 1-absorbing prime submodule, if for each and nonunits implies that or . 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 of a (faithfully) flat -module and any -module Also, we prove that if every proper submodule of an -module is weakly classical 1-absorbing prime, then . In terms of this result, we characterize modules over local rings in which every proper submodule is weakly classical 1-absorbing prime.
Paper Structure (2 sections, 25 theorems)

This paper contains 2 sections, 25 theorems.

Table of Contents

  1. Introduction
  2. Characterization of weakly classical 1-absorbing prime submodules

Key Result

Theorem 1

Let $M$ be an $R$-module and $N$ a proper submodule of $M$. If $(N:_{R}m)$ is a weakly 1-absorbing prime ideal of $R$ for each $m\in M\diagdown N,\ $then $N$ is a weakly classical 1-absorbing prime submodule of $M.$ The converse is true provided that $Ann_{R}(m)=0.$

Theorems & Definitions (56)

  • Definition 1
  • Example 1
  • Theorem 1
  • proof
  • Example 2
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Theorem 3
  • ...and 46 more