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Some factorization properties of idealization in commutative rings with zero divisors

Sina Eftekhari, Sayyed Heidar Jafari, Mahdi Reza Khorsandi

Abstract

We study some factorization properties of the idealization $R \mathop{(\! + \! )} M$ of a module $M$ in a commutative ring $R$ which is not necessarily a domain. We show that $R \mathop{(\! + \! )} M$ is ACCP if and only if $R$ is ACCP and $M$ satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which $R \mathop{(\! + \! )} M$ is a BFR. We also characterize the idealization rings which are UFRs.

Some factorization properties of idealization in commutative rings with zero divisors

Abstract

We study some factorization properties of the idealization of a module in a commutative ring which is not necessarily a domain. We show that is ACCP if and only if is ACCP and satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which is a BFR. We also characterize the idealization rings which are UFRs.

Paper Structure

This paper contains 2 sections, 7 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Results

Key Result

Lemma 1

Let $M$ be an $R$-module which is ACCC and let N be an $R$-submodule of $M$. Also, suppose that $R\overline{x_1} \subseteq R\overline{x_2} \subseteq \dotsb$ is an ascending chain of cyclic submodules in $M/N$, and $\overline{x_n} = r_{n}\overline{x_{n+1}}$ for some $r_n \in R$. If $N = r_iN$ for eve

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • Example 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • ...and 4 more