On constructions similar to the Burnside ring for commutative rings and profinite groups

Cornelius Greither and David Kent Harrison

Vol. 138 (1989), No. 1, 57–71

DOI: 10.2140/pjm.1989.138.57

Abstract

The question of finding all isomorphism classes of finite dimensional commutative semisimple rational algebras is an unsolved one and is equivalent to the question of finding all number fields. We feel that this problem may eventually be solved by the Burnside ring method, where the number fields are related to each other in many different ways. In this note we generalize the problem to the larger setting of G-algebras, where G is a finite abelian group. This gives even more relations—which we investigate. In order to see what is special about the rationals, we work as long as possible with a commutative ring R.

Mathematical Subject Classification 2000

Primary: 13B05

Secondary: 11E81

Milestones

Received: 28 October 1987

Revised: 3 May 1988

Published: 1 May 1989