When do pseudo-Gorenstein rings become Gorenstein?

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 pseudo-Gorenstein nearly Gorenstein graded domain 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.