Abstract
We will study modules of the highest injective, projective and flat dimension over a Goresntein ring. Let R be a Gorenstein ring of self-injective dimension n and 0 → RR → E0 → · · · → En → 0 a minimal injective resolution. Then it is shown in [F-I] that the flat dimension and projective dimension of En is n, the highest dimension. In this note, we shall prove that if M is a left R-module of injective dimension n, then the last injective term En(M) in a minimal injective resolution of M has projective and flat dimension n, and any indecomposable summand of En(M) embeds in En. As a consequence, we obtain that if R is Auslander-Gorenstein, then En(M) has essential socle.
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