Realizing orders as group rings

Abstract

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a universal grading, to answer questions about realizing orders as group rings. In particular, we address the Isomorphism Problem for group rings in the case where the ring is a reduced order. We prove that any non-zero reduced order R can be written as a group ring in a unique “maximal” way, up to isomorphism. More precisely, there exist a ring A and a finite abelian group G, both uniquely determined up to isomorphism, such that R≅A[G] as rings, and such that if B is a ring and H is a group, then R≅B[H] as rings if and only if there is a finite abelian group J such that B≅A[J] as rings and J×H≅G as groups. Computing A and G for given R can be done by means of an algorithm that is not quite polynomial-time. We also give a description of the automorphism group of R in terms of A and G.