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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 $M\in \mod R^{op}$ is a Gorenstein transpose of a module $A\in \mod R$ 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.