On Group Identities for the Unit Group of Algebras and Semigroup Algebras over an Infinite Field

Ann Dooms, Eric Jespers, Stanley Orlando Juriaans

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

Abstract

Let S be a semigroup generated by periodic elements and k be an infinite field. Suppose that the
semigroup algebra kS has 1. The authors prove interesting properties of kS and S assuming that
U(kS) satisfies a group identity (kS is called a GI-ring for short). In particular, S is locally finite
and kS satisfies a polynomial identity. Thus Hartley's conjecture is confirmed for such semigroup
algebras. If, furthermore, S is generated by a finite number of periodic elements, then they give
necessary and sufficient conditions on S for kS to be a GI-ring. The authors obtain this result by
proving first that if A is a semiprime GI-algebra over an infinite commutative domain D, such that
A is generated as a D-algebra by U(A) and no element of D is a zero divisor in A, then A is a
reduced ring. The latter fact is also used to give shorter proofs of some known results on group
rings which are GI-rings.
Original languageEnglish
Pages (from-to)273-283
Number of pages11
JournalJournal of Algebra
Volume284
Publication statusPublished - 2005

Bibliographical note

Journal of Algebra 284 no. 1 (2005), 273--283.

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