Gorenstein dimensions over some rings of the form \(R\otimes C\) (Q2796997)

Gorenstein projective, Gorenstein injective, and Gorenstein flat modules were introduced in [\textit{E. E. Enochs} and \textit{O. M. G. Jenda}, Math. Z. 220, No. 4, 611--633 (1995; Zbl 0845.16005)] and [\textit{E. E. Enochs} et al., J. Nanjing Univ., Math. Biq. 10, No. 1, 1--9 (1993; Zbl 0794.16001)]; these types of modules are the foundation of what is commonly known as Gorenstein homological algebra.NEWLINENEWLINEIn the paper [\textit{H. Holm} and \textit{P. Jørgensen}, J. Pure Appl. Algebra 205, No. 2, 423--445 (2006; Zbl 1094.13021)] a relative version of Gorenstein homological with respect to a so-called semidualizing module were developed. More precisely, for a semidualizing \(R\)-module \(C\) and any \(R\)-module \(M\) the authors define the \(C\)-Gorenstein projective, the \(C\)-Gorenstein injective, and the \(C\)-Gorenstein flat dimensions, denoted by \(C\text{-Gpd}_RM\), \(C\text{-Gid}_RM\), and \(C\text{-Gfd}_RM\), respectively. For \(C=R\) these dimensions agree with the (classic/absolute) Gorenstein projective, Gorenstein injective, and Gorenstein flat dimensions, \(\text{Gpd}_RM\), \(\text{Gid}_RM\), and \(\text{Gfd}_RM\), and in general one has NEWLINE\[NEWLINE C\text{-Gpd}_RM = \text{Gpd}_{R \ltimes C}M \; , \quad C\text{-Gid}_RM = \text{Gid}_{R \ltimes C}M \; , \quad \text{and} \quad C\text{-Gfd}_RM = \text{Gfd}_{R \ltimes C}M \; , NEWLINE\]NEWLINE where \(R \ltimes C\) is the trivial extension of \(R\) by \(C\).NEWLINENEWLINEThe main result in the paper under review is that the identities above also hold if one replaces the trivial extension \(R \ltimes C\) by other types of rings constructed from \(R\) and \(C\), such as e.g.~the amalgamated dublication \(R \bowtie C\) of \(R\) along \(C\).