Elham Tavasoli, Maryam Salimi, and Abolfazl Tehranian Islamic Azad University, Islamic Azad University, Islamic Azad University
Abstract : Let $R$ be a commutative Noetherian ring and let $I$ be an ideal of $R$. In this paper we study the amalgamated duplication ring $R\bowtie I$ which is introduced by D'Anna and Fontana. It is shown that if $R$ is generically Cohen-Macaulay $($resp. generically Gorenstein$)$ and $I$ is generically maximal Cohen-Macaulay $($resp. generically canonical module$)$, then $R\bowtie I$ is generically Cohen-Macaulay $($resp. generically Gorenstein$)$. We also defined generically quasi-Gorenstein ring and we investigate when $R\bowtie I$ is generically quasi-Gorenstein. In addition, it is shown that $R\bowtie I$ is approximately Cohen-Macaulay if and only if $R$ is approximately Cohen-Macaulay, provided some special conditions. Finally it is shown that if $R$ is approximately Gorenstein, then $R\bowtie I$ is approximately Gorenstein.
Keywords : amalgamated duplication, generically Cohen-Macaulay, generically Gorenstein, approximately Cohen-Macaulay, approximately Gorenstein
