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

TL;DR

This paper explores how certain ring-theoretic properties are transferred between a commutative ring and its associated Anderson ring, providing new insights and examples in the context of polynomial ring quotients.

Contribution

It introduces new results on the transfer of properties between a ring and its Anderson ring, expanding understanding of their algebraic relationship.

Findings

Established conditions for property transfer from R to R[X]_A

Provided examples illustrating property transfer and failure

Applied results to specific classes of rings

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.