Nonmatrix Varieties and Nil-Generated Algebras Whose Units Satisfy a Group Identity
✍ Scribed by Yuly Billig; David Riley; Vladimir Tasić
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 209 KB
- Volume
- 190
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
Let R = 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 nilpotent elements, then R = satisfies a group identity precisely when R satisfies a nonmatrix polynomial identity. As an application, we examine the group algebra FG of a torsion group G Ž . and the restricted enveloping algebra u L of a p-nil restricted Lie algebra L. Ž . = Giambruno, Sehgal, and Valenti recently proved that if the group of units FG satisfies a group identity, then FG satisfies a polynomial identity, thus confirming a Ž . = conjecture of Brian Hartley. We show that, in fact, FG satisfies a group identity if and only if FG satisfies a nonmatrix polynomial identity. In the case of restricted Ž . = enveloping algebras, we prove that u L satisfies a group identity if and only if Ž . u L satisfies the Engel condition.