On some algorithmic problems related to varieties of nonassociative rings (Q2713996)

There are two basic results in the article. In the first theorem, the author actually constructs a sequence of families of identities \((F_i)_{i\in N}\) so that 1) there is an algorithm that, given a natural \(n\), answers whether an arbitrary identity is in \(F_n\) and 2) there is no algorithm that, given a natural \(n\), answers whether \(F_n\) is equivalent to a finite family of identities. In the second theorem, the author constructs an infinite sequence of finitely based nonassociative rings \(A_1\supset B_1\supset A_2\supset B_2\supset\cdots\) so that the equational theory of \(A_i\) is undecidable and the equational theory of \(B_i\) is decidable, for all \(i\).