Minimal Reversible Nonsymmetric Rings

TL;DR

This paper proves that the smallest reversible nonsymmetric rings have a cardinality of 256 and provides examples of both duo and nonduo minimal rings, expanding understanding of their structure.

Contribution

It establishes the minimal size of reversible nonsymmetric rings and presents new examples, including nonduo rings, that were previously unknown.

Findings

Minimal reversible nonsymmetric rings have size 256.

$ ext{F}_2Q_8$ is a duo ring but minimal nonduo examples exist.

The work broadens the classification of minimal reversible nonsymmetric rings.

Abstract

Marks showed that $\mathbb{F}_2Q_8$, the $\mathbb{F}_2$ group algebra over the quaternion group, is a reversible nonsymmetric ring, then questioned whether or not this ring is minimal with respect to cardinality. In this work, it is shown that the cardinality of a minimal reversible nonsymmetric ring is indeed 256. Furthermore, it is shown that although $\mathbb{F}_2Q_8$ is a duo ring, there are also examples of minimal reversible nonsymmetric rings which are nonduo.