On decidability of the equational theories of ring varieties of finite characteristic

It is proved that for every natural number $n>1$ there exists a finitely based variety $\mathfrak X_n$ of (not necessarily associative) rings such that $\mathfrak X_n\models nx=0$, $\mathfrak X_n\not\models mx=0$ for every natural number $m<n$, and the equational theory $\mathfrak X_n$ is undecidable.