Nonmatrix Varieties and Nil-Generated Algebras Whose Units Satisfy a Group Identity Original Research Article

LetR×denote the group of units of an associative algebraRover an infinite fieldF. We prove that ifRis unitarily generated by its nilpotent elements, thenR×satisfies a group identity precisely whenRsatisfies a nonmatrix polynomial identity. As an application, we examine the group algebraFGof a torsion groupGand the restricted enveloping algebrau(L) of ap-nil restricted Lie algebraL. Giambruno, Sehgal, and Valenti recently proved that if the group of units (FG)×satisfies a group identity, thenFGsatisfies a polynomial identity, thus confirming a conjecture of Brian Hartley. We show that, in fact, (FG)×satisfies a group identity if and only ifFGsatisfies a nonmatrix polynomial identity. In the case of restricted enveloping algebras, we prove thatu(L)×satisfies a group identity if and only ifu(L) satisfies the Engel condition.