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 eprints
 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)
 EPrint:
 11 pages