The Socle of the Last Term in the Minimal Injective Resolution of a Gorenstein Module

Abstract

Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and $_RU$ a Gorenstein module with $S={\rm End}(_RU)$. If the injective dimensions of $_RU$ and $U_S$ are finite, then the last term in the minimal injective resolution of $_{R}U$ has an essential socle.