A Cayley-Hamilton trace identity for 2 x 2 matrices over Lie-solvable rings

TL;DR

This paper presents a Cayley-Hamilton trace identity specifically for 2x2 matrices over rings that satisfy a certain Lie-solvability condition and contain 1/2, extending classical matrix identities to a broader algebraic context.

Contribution

It introduces a new Cayley-Hamilton trace identity for 2x2 matrices over Lie-solvable rings with 1/2, expanding the understanding of matrix identities in non-commutative rings.

Findings

Derived a Cayley-Hamilton trace identity for matrices over Lie-solvable rings.

Established conditions under which the identity holds, including the presence of 1/2 in the ring.

Extended classical matrix identities to a broader class of rings.

Abstract

We exhibit a Cayley-Hamilton trace identity for $2\times2$ matrices with entries in a ring $R$ satisfying $[[x,y],[x,z]]=0$ and 1/2 \in R$.