Characterization of almost Gorenstein rings in terms of the trace ideal

TL;DR

The paper develops a trace-ideal criterion for almost Gorensteinness in dimension one, showing $R$ is almost Gorenstein if and only if $\mathrm{tr}_R(\mathfrak{m}\mathrm{K}_R) \supseteq \mathfrak{m}$, and extends this lens to analyze $\mathbb{Z}_2$-graded rings of the form $A=R\times_{\varphi}\mathfrak{m}$ via a triad $(R,\mathfrak{m},\varphi)$ with $\varphi$ encoded by $\alpha \in \mathfrak{m}:\mathfrak{m}^2$. It proves an equivalence for $A$ being almost Gorenstein in terms of $R$ and the trace of $\langle 1,\alpha\rangle_B$ over $B=\mathfrak{m}:\mathfrak{m}$, and it provides a comprehensive structure theory for $\mathbb{Z}_2$-graded rings, including Gorenstein, CM, and regularity properties, with detailed appendices on Gorenstein criteria and regularity. The results unify idealization and graded-ring constructions, yielding practical criteria and explicit constructions in low dimensions and offering tools for broader graded-ring analysis.

Abstract

We provide a characterization of one-dimensional almost Gorenstein rings in terms of the trace ideal. As an application, we investigate the almost Gorenstein property of certain $\mathbb{Z}_2$-graded rings.

Theorems & Definitions (98)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.4
• Lemma 2.5
• proof
• Theorem 2.7
• proof
• Remark 2.8
• Remark 2.9