# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 129, 2013, pp. 115–127**

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

The Amalgamated Duplication of a Ring Along a Semidualizing Ideal

Maryam Salimi^{[1]}, Elham Tavasoli

^{[2]}and S. Yassemi

^{[3]}(1) Department of Mathematics, Islamic Azad University, Tehran, Iran

(2) Department of Mathematics, Islamic Azad University, Tehran, Iran

(3) School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, 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