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Let $G$ be a group and let $k$ be a field. Kaplansky's direct finitenessconjecture states that every one-sided unit of the group ring $k[G]$ must be atwo-sided unit. In this paper, we establish a geometric direct finitenesstheorem for endomorphisms of symbolic algebraic varieties. Whenever $G$ is asofic group or more generally a surjunctive group, our result implies ageneralization of Kaplansky's direct finiteness conjecture for the near ring$R(k, G)$ which is $k[X_g\colon g \in G]$ as a group and which containsnaturally $k[G]$ as the subring of homogeneous polynomials of degree one. Wealso prove that Kaplansky's stable finiteness conjecture is a consequence ofGottschalk's Surjunctivity conjecture.

A geometric generalization of Kaplansky's direct finiteness conjecture