# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 129, 2013, pp. 115–127**

**DOI: 10.4171/RSMUP/129-8**

Published online: 2013-05-15

The Amalgamated Duplication of a Ring Along a Semidualizing Ideal

Maryam Salimi^{[1]}, Elham Tavasoli

^{[2]}and S. Yassemi

^{[3]}(1) Islamic Azad University, Teheran, Iran

(2) Islamic Azad University, Teheran, Iran

(3) Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

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) = {\rm fd} _{R \bowtie I} (M \otimes _{R} (R\bowtie I)) = {\rm fd} _{R} (M\otimes _{R}(R\bowtie I))$, for every $R$-module $M$.

*Keywords: *Amalgamated duplication, semidualizing, totally reflexive, Gorenstein projective

Salimi Maryam, Tavasoli Elham, Yassemi S.: The Amalgamated Duplication of a Ring Along a Semidualizing Ideal. *Rend. Sem. Mat. Univ. Padova* 129 (2013), 115-127. doi: 10.4171/RSMUP/129-8