The reduction number of canonical ideals

TL;DR

This paper introduces a new invariant based on the reduction number of canonical ideals in Cohen-Macaulay rings, helping to measure how close a ring is to being Gorenstein and exploring related properties.

Contribution

It defines a novel invariant applicable to arbitrary Cohen-Macaulay rings and links it to almost Gorenstein and nearly Gorenstein properties, as well as idealizations.

Findings

Relates the invariant to almost Gorenstein and nearly Gorenstein rings in dimension one.

Characterizes idealizations of trace ideals over Gorenstein rings using the invariant.

Provides insights into the Gorenstein proximity of Cohen-Macaulay rings.

Abstract

In this paper, we introduce an invariant of Cohen-Macaulay local rings in terms of the reduction number of canonical ideals. The invariant can be defined in arbitrary Cohen-Macaulay rings and it measures how close to being Gorenstein. First, we clarify the relation between almost Gorenstein rings and nearly Gorenstein rings by using the invariant in dimension one. We next characterize the idealization of trace ideals over Gorenstein rings in terms of the invariant. It provides better prospects for a result of the almost Gorenstein property of idealization.