On generalized Gorenstein local rings

TL;DR

This paper introduces generalized Gorenstein local rings, extending the concept of almost Gorenstein rings, and explores their properties through various algebraic structures and examples.

Contribution

It defines GGL rings as a natural extension of almost Gorenstein rings and investigates their properties via endomorphism algebras, trace ideals, Ulrich ideals, and Rees algebras.

Findings

GGL rings generalize almost Gorenstein rings.

Connections between GGL property and various algebraic structures.

Examples include numerical semigroup rings and determinantal rings.

Abstract

In this paper, we introduce generalized Gorenstein local (GGL) rings. The notion of GGL rings is a natural generalization of the notion of almost Gorenstein rings, which can thus be treated as part of the theory of GGL rings. For a Cohen-Macaulay local ring $R$, we explore the endomorphism algebra of the maximal ideal, the trace ideal of the canonical module, Ulrich ideals, and Rees algebras of parameter ideals in connection with the GGL property. We also give numerous examples of numerical semigroup rings, idealizations, and determinantal rings of certain matrices.