Stable finiteness of monoid algebras and surjunctivity

A monoid $M$ is said to be surjunctive if every injective cellular automaton with finite alphabet over $M$ is surjective. We show that monoid algebras of surjunctive monoids are stably finite. In other words, given any field $K$ and any surjunctive monoid $M$, every one-sided invertible square matrix with entries in the monoid algebra $K[M]$ is two-sided invertible. Our proof uses first-order model theory.