NONMATRIX VARIETIES AND NIL-GENERATED ALGEBRAS WHOSE UNITS SATISFY A GROUP IDENTITY

Let R-x denote the group of units of an associative algebra R over an infinite field F. We prove that if R is unitarily generated by its nil potent elements, then R-x satisfies a group identity precisely when R satisfies a nonmatrix polynomial identity. As an application, we exami ne the group algebra FG of a torsion group G and the restricted envelo ping algebra u(L) of a p-nil restricted Lie algebra L. Giambruno, Sehg al, and Valenti recently proved that if the group of units (FC)(x) sat isfies a group identity, then FG satisfies a polynomial identity, thus confirming a conjecture of Brian Hartley. We show that, in fact, (FG) (x) satisfies a group identity if and only if FG satisfies a nonmatrix polynomial identity. In the case of restricted enveloping algebras, w e prove that u(L)(x) satisfies a group identity if and only if u(L) sa tisfies the Engel condition. (C) 1997 Academic Press.