Realizing orders as group rings

Journal of Algebra, vol. 641, pp. 391–428

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.

Authors 3

1. Hendrik W. Lenstra corresponding Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, the Netherlands

2. A. Silverberg corresponding

   University of California, Irvine

   Affiliation as printed

   Department of Mathematics, University of California, Irvine, CA 92697, USA

3. Daniël van Gent corresponding Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, the Netherlands