Universal equivalence of general linear groups over local rings with 1/2

TL;DR

This paper proves that for local rings with 1/2, the universal equivalence of general linear groups with inverse-transpose automorphism is equivalent to the rings having the same order and being universally equivalent.

Contribution

It establishes a precise criterion linking the universal equivalence of certain linear groups to the rings' properties over local rings with 1/2.

Findings

Universal equivalence of GL groups implies rings have same order

Universal equivalence of rings corresponds to group equivalence

Results hold for non-commutative local rings with 1/2

Abstract

In this study, it is proven that the universal equivalence of general linear groups (admitting the inverse-transpose automorphism) of orders greater than $2$, over local, not necessarily commutative rings with $1/2$, is equivalent to the coincidence of the orders of the groups and the universal equivalence of the corresponding rings.