On canonical modules of idealizations

Abstract

Let $(R,\mathfrak {m})$ be a Noetherian local ring which is a quotient of a Gorenstein local ring. Let $M$ be a finitely generated $R$-module. In this paper, we study the structure of the canonical module $K(R\ \mathbb {n}\ M)$ of the idealization $R\ \mathbb {n}\ M$ via the polynomial type introduced by Cuong~\cite {C}. In particular, we give a characterization for $K(R\ \mathbb {n}\ M)$ being Cohen-Macaulay and generalized Cohen-Macaulay.