A LOCALLY FINITE VARIETY OF RINGS WITH AN UNDECIDABLE EQUATIONAL THEORY

We prove that the non-finitely based system of polynomial identities over an arbitrary field ${\mathbb{F}}$ of characteristic 2 given by Gupta and Krasilnikov in [A non-finitely based system of polynomial identities which contains the identity x⁶ = 0, Quart. J. Math. 53 (2002), 173–183] can, by a slight modification, be turned into an independent system. This has important consequences for algorithmic problems in algebra: there is a locally finite-dimensional variety of associative algebras (and, in particular, a locally finite variety of rings) that has an undecidable equational theory. For such a variety, the uniform word problem is unsolvable, and yet the word problem is recursively solvable for each individual finitely presented algebra (ring) in that variety.