A Conjecture Relating to the Isomorphism Problem for Commutative Group Algebras

Abstract

This chapter describes a conjecture concerning the isomorphism problem for commutative group algebras. Suppose R is a commutative ring with identity, and let G and H be abelian groups (possibly infinite). Let GR (respectively HR) be the direct sum of the p-components of torsion in g (respectively h) such that the prime p inverts in R. It was shown earlier that if R satisfies the isomorphism theorem, then it is a nicely decomposing ring in the following sense: Let inv(R) denote the set of prime numbers that invert in R. Suppose R is a finite product of indecomposable rings, each of characteristic O. Then, R satisfies the isomorphism theorem only if R is an ND-ring.