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The amalgamated duplication of a ring along a semidualizing ideal. (English) Zbl 1279.13025
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 M. D’Anna and M. Fontana [J. Algebra Appl. 6, No. 3, 443–459 (2007; Zbl 1126.13002)], 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 \(\text{fd}_{R}(M) = \text{fd} _{R \bowtie I} (M \otimes _{R} (R\bowtie I)) = \text{fd} _{R} (M\otimes _{R}(R\bowtie I))\), for every \(R\)-module \(M\).

13D05 Homological dimension and commutative rings
Full Text: DOI
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