A linear variant of the nearly Gorenstein property

The nearly Gorenstein property is a generalization of the Gorenstein property. In this paper, we introduce a new property () that is broader than the nearly Gorenstein property in Cohen-Macaulay semi-standard graded rings. We show that the property () is related to a property that has been studied in affine semigroup rings. We generalize some results of Veronese subrings from the aspect of the property (). Moreover, we prove that the property () and the nearly Gorenstein property are equivalent in both the Ehrhart rings of $(0,1)$-polytopes with IDP and rings which have minimal multiplicity.