Abstract:
It is proven that there exists no algorithm deciding whether the variety $\mathrm{var}\Sigma$ is finitely based relative to an arbitrary recursive system of ring identities $\Sigma$. An infinite sequence is constructed of finitely based varieties of nonassociative rings $\mathfrak A_1\supset\mathfrak B_1\supset\mathfrak A_2\supset\mathfrak B_2 \supset\dotsb$ such that, for all $i$, the equational theory of $\mathfrak A_i$ is undecidable and the equational theory of $\mathfrak B_i$ is decidable.
Key words:variety of rings, finitely based variety, equational theory, decidable theory, undecidable theory.
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