A solution of a problem of Plotkin and Vovsi and an application to varieties of groups

Let K be an arbitrary field of characteristic 2, F a free group of countabl y infinite rank. We construct a finitely generated fully invariant subgroup U in F such that the relatively free group F/U satisfies the maximal condi tion on fully invariant subgroups but the group algebra K(F/U) does not sat isfy the maximal condition on fully invariant ideals. This solves a problem posed by Plotkin and Vovsi. Using the developed techniques we also constru ct the first example of a non-finitely based (nilpotent of class 2)-by-(nil potent of class 2) variety whose Abelian-by-(nilpotent of class at most 2) groups form a hereditarily finitely based subvariety.