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

Let $R$ be a commutative ring with unity and let $X$ be an indeterminate over $R$. The $\mathit{\text{Anderson}}\mathit{\text{ring}}$ 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.