Lattices of varieties of algebras

Let $A$ be an associative and commutative ring with 1, $S$ a subsemigroup of the multiplicative semigroup of $A$, not containing divisors of zero, and $\mathfrak X$ some variety of $A$-algebras. A study is made of the homomorphism from the lattice $L(\mathfrak X)$ of all subvarieties of $\mathfrak X$ into the latttice of all varieties of $S^{-1}A$-algebras, which is induced in a certain natural sense by the functor $S^{-1}$. Under one weak restriction on $\mathfrak X$ a description is given of the kernel of this homomorphism, and this makes it possible to establish a good interrelation between the properties of the lattice $L(\mathfrak X)$ and the lattice of varieties of $S^{-1}A$-algebras. These results are applied to prove that a number of varieties of associative and Lie rings have the Specht property.
Bibliography: 18 titles.