Associative algebras satisfying a semigroup identity
Abstract
Denote by (R,.) the multiplicative semigroup of an associative algebra R over an infinite field, and let (R,*) represent R when viewed as a semigroup via the circle operation x*y=x+y+xy. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of R. Namely, we prove that the following conditions on R are equivalent: the semigroup (R,*) satisfies an identity; the semigroup (R,.) satisfies a reduced identity; and, the associated Lie algebra of R satisfies the Engel condition. When R is finitely generated these conditions are each equivalent to R being upper Lie nilpotent.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- February 1998
- DOI:
- 10.48550/arXiv.math/9802039
- arXiv:
- arXiv:math/9802039
- Bibcode:
- 1998math......2039R
- Keywords:
-
- Rings and Algebras;
- 16R40 (Primary) 20M07;
- 20M25 (Secondary)
- E-Print:
- 11 pages