Square-Free Rings And Their Automorphism Group

TL;DR

This paper characterizes square-free rings as twisted semigroup rings over square-free semigroups with coefficients in division rings, and explores their automorphism groups via cohomology, extending previous classifications.

Contribution

It provides a new characterization of square-free rings over division rings and links their automorphisms to cohomological structures, generalizing earlier results.

Findings

Square-free rings are characterized as twisted semigroup rings over division rings.

A short exact sequence relates automorphisms of square-free rings to cohomology groups.

The work extends classification from semigroup algebras to more general rings.

Abstract

Finite-dimensional square-free algebras have been completely characterized by Anderson and D'Ambrosia as certain twisted semigroup algebras over a square-free semigroup S with coefficients in a field K. D'Ambrosia extended the definition of square-free to artinian rings with unity and showed every square-free ring has an associated division ring D and square-free semigroup S. We show a square-free ring can be characterized as a twisted semigroup ring over a square-free semigroup S with coefficients in a division ring D. Also, to each square-free ring there exists a short exact sequence connecting the outer automorphisms of a square-free ring to certain cohomology groups related to S and D.