Gorenstein Dimensions over Some Rings of the Form \(R \oplus C\)

Given a semidualizing module \(C\) over a commutative noetherian ring, Holm and J\o{}rgensen investigate some connections between \(C\)-Gorenstein dimensions of an \(R\)-complex and Gorenstein dimensions of the same complex viewed as a complex over the "trivial extension" \(R \ltimes C\). We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D'Anna and Fontana's amalgamated duplication and Enescu's pseudocanonical cover.