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On the transfer of certain ring-theoretic properties in Anderson rings

Hyungtae Baek, Jung Wook Lim, Ali Tamoussit

Abstract

Let $R$ be a commutative ring with unity and let $X$ be an indeterminate over $R$. The \textit{Anderson ring} of $R$ is defined as the quotient ring of the polynomial ring $R[X]$ by the set of polynomials that evaluate to $1$ at $0$. Specifically, the Anderson ring of $R$ is $R[X]_A$, where $A=\{f\in R[X]\mid f(0)=1\}$. In this paper, we aim to investigate the transfer of various ring-theoretic properties between the ring $R$ and its Anderson ring $R[X]_A$. Interesting results are established, accompanied by applications and illustrative examples.

On the transfer of certain ring-theoretic properties in Anderson rings

Abstract

Let be a commutative ring with unity and let be an indeterminate over . The \textit{Anderson ring} of is defined as the quotient ring of the polynomial ring by the set of polynomials that evaluate to at . Specifically, the Anderson ring of is , where . In this paper, we aim to investigate the transfer of various ring-theoretic properties between the ring and its Anderson ring . Interesting results are established, accompanied by applications and illustrative examples.

Paper Structure

This paper contains 6 sections, 31 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Specific ring-theoretic properties of Anderson rings
  3. Jaffard-like domains arising from Anderson rings
  4. Specific integral domains related to star-operations arising from Anderson rings
  5. Preliminaries: definitions and notation
  6. Main results and applications

Key Result

Theorem 2.2

Let $R$ be a commutative ring with identity. The following assertions hold.

Theorems & Definitions (64)

  • Remark 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 54 more