A power Cayley-Hamilton identity for nxn matrices over a Lie nilpotent ring of index k

TL;DR

This paper establishes a generalized Cayley-Hamilton identity for matrices over Lie nilpotent rings, revealing new algebraic invariants and polynomial identities that depend on the ring's nilpotency index.

Contribution

It introduces a power Cayley-Hamilton identity of degree (n^2)2^{k-2} for matrices over Lie nilpotent rings, extending classical results to noncommutative settings.

Findings

Proves a new Cayley-Hamilton identity for Lie nilpotent rings.

Shows the coefficients are not uniquely determined by the matrix.

Identifies the role of the double commutator ideal in the characteristic polynomial.

Abstract

For an nxn matrix A over a Lie nilpotent ring R of index k, we prove that an invariant "power" Cayley-Hamilton identity of degree (n^2)2^{k-2} holds. The right coefficients are not uniquely determined by A, and the cosets lambda_i+D, with D the double commutator ideal R[[R,R],R]R of R, appear in the so-called second right characteristic polynomial of the natural image of A in the nxn matrix ring M_{n}(R/D) over the factor ring R/D.