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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.

Author

Riley, David M.; Wilson, Mark C.

Published

1998

Language

EN