Two theorems about equationally noetherian groups

An algebraic set over a group G is the set of all solutions of some system {f(x₁,...,x_n) = 1 | f ∈ G *>x₁,...,x_n<} of equations over G. A group G is equationally noetherian if every algebraic set over G is the set of all solutions of a finite subsystem of the given one. We prove that a virtually equationally noetherian group is equationally noetherian and that the quotient of an equationally noetherian group by a normal subgroup which is a finite union of algebraic sets is again equationally noetherian. On the other hand, the wreath product W = U wreath product T of a non-abelian group U and an infinite group T is not equationally noetherian.