Invariant ideals of abelian group algebras under the multiplicative action of a field. I

• Published in: Proceedings of the American Mathematical Society Vol. 130; no. 4; pp. 939 - 949
• Main Authors: Passman, D. S., Zalesskiǐ, A. E.
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-04-2002
• Subjects: Algebra; Associative rings and algebras; College mathematics; Exact sciences and technology; Field theory and polynomials; Mathematical induction; Mathematical rings; Mathematical theorems; Mathematics; Sciences and techniques of general use; Uniqueness; Written composition; Invariant; Group algebra; Vector space; Abelian group; Ideal (mathematics); Group theory; Invariant ideal; Finite dimension; Division ring; Multiplicative action
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-01-06092-0

Let D be a division ring and let V=D^n be a finite-dimensional D-vector space, viewed multiplicatively. If G=D^\bullet is the multiplicative group of D, then G acts on V and hence on any group algebra K[V]. Our goal is to completely describe the semiprime G-stable ideals of K[V]. As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields. Part I of this work is concerned with the latter situation, while Part II deals with arbitrary division rings.