# Classify all Rings with every unit having order dividing 24

I'm working on the following qual question:

Suppose that $R$ is a finite ring with 1 such that every unit of $R$ has order dividing 24. Classify all such $R$.

My thoughts are I would like to try to show that there is an $x$ such that $axa = a$ for all $a$ in the ring. So maybe that can tell me the ring is von Neumann regular and then I can use that somehow. The other challenge is that we are only given information on the units, and we don't know much about the nonunits. Any guidance would be much appreciated

Source: Jan 2017

• If $n$ is the characteristic of $R$ then any prime factor of $n$ must be one of $2, 3, 5, 7, 13$. – Kenny Lau Aug 11 '18 at 8:34
• Start by finding the possible characteristics. – Tobias Kildetoft Aug 11 '18 at 8:35
• Can the characteristic be composite? – iYOA Aug 11 '18 at 8:46
• Multiplicative order or...?! – rschwieb Aug 13 '18 at 20:28
• $F_2[x]/(x^2)$ has units of multiplicative order $1$ and $2$, and it should put any idea of proving the ring is VNR to rest. Incidentally everything has additive order $2$ if that was meant instead. – rschwieb Aug 13 '18 at 20:32