Identities of PI-Algebras Graded by a Finite Abelian Group

Abstract

We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ℤ₂)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.

Key Words:

• Associative algebras
• Graded identities
• Group-graded algebras
• Identities with automorphisms
• PI-algebras

2000 Mathematics Subject Classification:

• Primary 16R50
• Secondary 16R10, 16W50, 16W22

Notes

After the submission of the present article, Eli Aljadeff and Alexei Kanel-Belov posted the preprint arXiv:0903.0362 (the latest version 0903.0362v3 from May 6, 2009) where they use similar methods and independently establish results similar to Theorems 1, 2, 3 in the more general situation of PI-algebras graded by an arbitrary finite group.

Communicated by I. Shestakov.