A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic

We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p, we obtain a field K of order p${}^4$, a finitely generated group G containing torsion, and finite sums $a,b\in K[G]$ with $ab=1$ but $ba\ne 1$. The same elements define a cellular automaton on K${}^G$ that is injective but not surjective.