On group identities for the unit group of algebras and semigroup algebras over an infinite field

Abstract

Let k be an infinite field. We fully describe when the unit group of a semigroup algebra k[S] of a semigroup S generated by finitely many periodic elements satisfies a group identity. This and some other recent results are proved by first showing that semiprime k-algebras generated by units are necessarily reduced whenever their unit group satisfies a group identity.  2004 Elsevier Inc. All rights reserved.The unit group U(R) of a ring R with unity 1 is said to satisfy a group identity (we call R a GI-ring for short) if there exists a non-trivial word w(x 1 , . . . , x n ) in the free group generated by x 1 , . . . , x n such that w(u 1 , . . . , u n ) = 1 for all u 1 , . . . , u n ∈ U(R).We start with a brief overview of some recent results on this topic. As a consequence of a more general result, Valitskas in [23] proved that if A is an algebra over an infinite field ✩ Research partially supported by the Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Vlaanderen), Flemish-Polish bilateral agreement BIL 01/31, FAPESP-Brazil and CNPq-Brazil (Proc. 300652/95-0).

Symmetric units satisfying a group identity

“…Since K is an infinite field, we have by [4,Corollary 1.4(2)] that GL n i (D i ) has to be commutative, thus n i = 1 and D i is a field K i .…”

Star-group identities and groups of units

“…Actually it turns out that one can completely characterize those groups G for which U(F G) satisfies a group identity [14,16]. Related results were obtained in the following years, see for instance [2,3,8].…”

Group identities on the units of algebraic algebras with applications to restricted enveloping algebras

“…See [8] for a recent and comprehensive overview. In particular, it was shown in [17] that every algebraic GI-algebra is locally finite.…”

“…Let A be an algebraic algebra over an infinite field F of characteristic p 0. Then the following conditions are equivalent: satisfies a group identity then Proposition 1.2 and Theorem 1.3 in [8] apply yielding the fact that N (A) is a locally nilpotent ideal of A coinciding with the prime radical B(A). By Theorem 1.2 of [6], the subalgebra F · 1 + N (A) satisfies a non-matrix polynomial identity.…”

“…By Theorem 1.2 of [6], the subalgebra F · 1 + N (A) satisfies a non-matrix polynomial identity. Since A/B(A) is commutative, by Corollary 1.4 of [8], it is easy to check that A itself satisfies a non-matrix polynomial identity (see [23]). The remaining implications follow as in Theorems 1.3 and 1.4 in [6].…”