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Algebras with Collapsing Monomials
Author(s) -
Riley David Michael
Publication year - 1998
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609398004597
Subject(s) - mathematics , semigroup , nilpotent , integer (computer science) , lie algebra , monomial , pure mathematics , unitary state , zero (linguistics) , associative property , field (mathematics) , combinatorics , discrete mathematics , algebra over a field , law , linguistics , philosophy , computer science , political science , programming language
A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S , at least two distinct words of length n on these letters are equal in S . In particular, S is collapsing whenever it satisfies a law. Let U ( A ) denote the group of units of a unitary associative algebra A over a field k of characteristic zero. If A is generated by its nilpotent elements, then the following conditions are equivalent: U ( A ) is collapsing; U ( A ) satisfies some semigroup law; U ( A ) satisfies the Engel condition; U ( A ) is nilpotent; A is nilpotent when considered as a Lie algebra.
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