Gorenstein syzygy modules

For any ring R and any positive integer n, we prove that a left R-module is a Gorenstein n-syzygy if and only if it is an n-syzygy. Over a left and right Noetherian ring, we introduce the notion of the Gorenstein transpose of finitely generated modules. We prove that a module is a Gorenstein transpose of a module if and only if M can be embedded into a transpose of A with the cokernel Gorenstein projective. Some applications of this result are given.