When do pseudo‐Gorenstein rings become Gorenstein?

Abstract We discuss the relationship between the trace ideal of the canonical module and pseudo‐Gorensteinness. In particular, under certain mild assumptions, we show that every positively graded domain that is both pseudo‐Gorenstein and nearly Gorenstein is Gorenstein. As an application, we clarify the relationships among nearly Gorensteinness, almost Gorensteinness, and levelness — notions that generalize Gorensteinness — in the context of standard graded domains. Moreover, we give a method for constructing quasi‐Gorenstein rings by taking a Veronese subalgebra of certain Noetherian graded rings.