Unipotent element of order a unit is trivial

Let $A$ be a ring, not assumed commutative, and let $u \in A$ be an element such that $u - 1$ is nilpotent, i.e. $(u-1)^k = 0$ for some $k$. Let $m$ be a natural number whose image $m \cdot 1_A$ under the canonical map $\mathbb{N} \to A$ is a unit of $A$. Assume $u^m = 1$. Then $u = 1$. No hypothesis of positivity on $m$ is imposed beyond the invertibility of its image (which in particular excludes $m = 0$ unless $A$ is trivial), and no finiteness or commutativity assumption is placed on $A$.

This is the elementary statement that a unipotent element of a ring has no torsion of order invertible in the ring — classically, that a unipotent matrix of finite order prime to the characteristic is the identity. It is used in the proof of Algebra.exists_pow_eq_one_and_forall_algHom_apply_eq_of_locally_of_isUnit_natCast.