The finite basis property for the union of some varieties of algebras (Q923699)

Let F be an associative and commutative ring with 1 and \({\mathfrak A}\) be the variety of all associative algebras over F. Assume \({\mathfrak B}\) is a variety (over F) with a finitely based T-ideal T(\({\mathfrak B})\) and there is a multilinear polynomial in T(\({\mathfrak B})\) whose sum of coefficients is an invertible element in F. The main theorem in the present note states that the union of the varieties \({\mathfrak A}\) and \({\mathfrak B}\) (i.e. the least variety that contains both \({\mathfrak A}\) and \({\mathfrak B})\) is finitely based. As a straightforward consequence of this theorem it is obtained that if \({\mathfrak B}\) is a finitely based variety of Lie algebras (1/2\(\in F)\) or of Mal'cev algebras then the union of \({\mathfrak A}\) and \({\mathfrak B}\) is finitely based as well.