The Amalgamated Duplication of a Ring Along a Semidualizing Ideal

Abstract

Let $R$ be a commutative Noetherian ring and let $I$ be an ideal of $R$. In this paper, after recalling briefly the main properties of the amalgamated duplication ring $R\bowtie I$ which is introduced by D'Anna and Fontana, we restrict our attention to the study of the properties of $R\bowtie I$, when $I$ is a semidualizing ideal of $R$, i.e., $I$ is an ideal of $R$ and $I$ is a semidualizing $R$-module. In particular, it is shown that if $I$ is a semidualizing ideal and $M$ is a finitely generated $R$-module, then $M$ is totally $I$-reflexive as an $R$-module if and only if $M$ is totally reflexive as an $(R\bowtie I)$-module. In addition, it is shown that if $I$ is a semidualizing ideal, then $R$ and $I$ are Gorenstein projective over $R\bowtie I$, and every injective $R$-module is Gorenstein injective as an $(R\bowtie I)$-module. Finally, it is proved that if $I$ is a non-zero flat ideal of $R$, then fd ${}_R(M)={\mathrm{fd}}_{R\bowtie I}(M{\otimes }_R(R\bowtie I))={\mathrm{fd}}_R(M{\otimes }_R(R\bowtie I))$, for every $R$-module $M$.