Gorenstein flat preenvelopes

Abstract

We consider a two sided noetherian ring $R$ such that the character modules of Gorenstein injective left $R$-modules are Gorenstein flat right $R$-modules. We then prove that the class of Gorenstein flat right $R$-modules is preenveloping. We also show that the class of Gorenstein flat complexes of right $R$-modules is preenevloping in $\mathit{Ch}(R)$. In the second part of the paper we give examples of rings with the property that the character modules of Gorenstein injective modules are Gorenstein flat. We prove that any two sided noetherian ring $R$ with $\mathop{\mathit{i.d.}}_{R^{\mathrm{op}}} R < \infty$ has the desired property. We also prove that if $R$ is a two sided noetherian ring with a dualizing bimodule ${}_{R}V_{R}$ and such that $R$ is left $n$-perfect for some positive integer $n$, then the character modules of Gorenstein injective modules are Gorenstein flat.