Finite unitary rings all of whose groups of units of all their subrings except of the ring itself are solvable

TL;DR

This paper classifies finite unitary rings where the group of units of the ring itself is non-solvable, but all proper subring unit groups are solvable, focusing on rings of small order.

Contribution

It provides a classification of such rings and identifies all finite rings of certain small orders that belong to this class.

Findings

All finite rings of order p^n for n<5 are in this class.

Some rings of order p^6 are also in this class.

The rings in this class have a non-solvable unit group only for the entire ring, not for its proper subrings.

Abstract

Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper we classify these rings and show that all finite rings of order $p^n$ for $n < 5$ and some of order $p^6$ are in this class of rings.