Trace Ideals and the Gorenstein Property

TL;DR

This paper characterizes Artinian Gorenstein rings through the property that all ideals are trace ideals, providing a new perspective on the structure of such rings.

Contribution

It establishes a characterization of Artinian Gorenstein rings via trace ideals, linking ideal properties to the Gorenstein condition.

Findings

R is Artinian Gorenstein iff every ideal is a trace ideal

Conditions under which trace ideals coincide with double annihilators

New insights into the structure of Gorenstein rings

Abstract

Let R be a local Noetherian commutative ring. We prove that R is an Artinian Gorenstein ring if and only if every ideal in R is a trace ideal. We discuss when the trace ideal of a module coincides with its double annihilator.