suppose first that are nilpotent and is a unit. then is nilpotent and thus is a unit becasue, in any commutative ring, nilpotent + unit is

a unit. for the converse, we'll use induction on the degree of : it's clear if n = 0. so suppose the claim is true for any polynomial which is a unit and has degree < n.

let be a unit. so there exists such that if then and is a unit. so we may assume

that m > 0. clearly must be a unit and we also have:

write these relations in terms of matrices: where

where means that we don't care what those entries are. the important point is that is a (lower) triangular matrix with

on its main diagonal. now multiply from the left, by the adjoint matrix of to get thus and hence because

is a unit. so and therefore is nilpotent. hence is a unit, because, as i already mentioned, nilpotent + unit is a unit. now since we can

apply the induction hypothesis to finish the proof.