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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 $$M$$.

##### MSC:
 13D05 Homological dimension and commutative rings
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