Abstract:
A variety of Lie rings is called just-non-Cross if it itself is not Cross, but each of its proper subvarieties is Cross, i.e. is generated by a finite ring. In this paper, we completely describe the solvable just-non-Cross varieties both of Lie rings and of Lie $R$-algebras where $R$ is a finite commutative ring with identity and, in particular, where $R$ is a finite field. We find algorithms which allow us to determine whether a given identity defines a Cross variety of Lie algebras; also, using the multiplication and addition tables of a finite Lie algebra, we find algorithms for extracting its identities.
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