1998/02/06 by David M. Riley, Mark C. Wilson, Riley, David M. +1
Materials Science · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Synthesis and properties of polymers #math.RA #msc:16R40 #msc:20M07 #msc:20M25
paper · pdf · doi:10.48550/arxiv.math/9802039
11 pages; written in LaTeX2e
arxiv created 1998/02/06 · arxiv updated 2009/11/30
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.