Graded Frobenius Rings

TL;DR

This paper introduces and studies graded Frobenius rings and related structures, providing new theoretical insights into their properties, structure, and representations from a ring-theoretic perspective.

Contribution

It defines graded quasi-Frobenius and Frobenius rings, explores their structure, and extends classical results to the graded setting, including shift versions.

Findings

Characterization of graded Frobenius rings

Structure theorem for shift-graded Frobenius rings

Connections between radicals and finiteness conditions in graded rings

Abstract

In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings.