C-Gorenstein projective, injective, flat modules and trivial ring extensions

Let C be a semidualizing module over a commutative ring R and let R & ltimes; C {R\ltimes C} be the trivial extension of R by C. We prove that (1) ( M , f ) {(M,f)} is a Gorenstein projective R & ltimes; C {R\ltimes C} -module if and only if M is a C-Gorenstein projective R-module; (2) [ M , g ] {[M,g]} is a Gorenstein injective R & ltimes; C {R\ltimes C} -module if and only if M is a C-Gorenstein injective R-module; (3) if R is a coherent ring, then ( M , f ) {(M,f)} is a Gorenstein flat R & ltimes; C {R\ltimes C} -module if and only if M is a C-Gorenstein flat R-module. Some applications of these results are given.