Realizing orders as group rings

TL;DR

This paper demonstrates that every non-zero reduced order can be uniquely expressed as a group ring, providing a solution to the Isomorphism Problem for such rings and describing their automorphism groups.

Contribution

It introduces the universal grading of reduced orders and proves their unique decomposition as a group ring, advancing understanding of their structure and automorphisms.

Findings

Every reduced order has a unique maximal group ring decomposition.

The decomposition is unique up to isomorphism of the ring and the group.

An algorithm for computing the decomposition exists, though not polynomial-time.

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 $R\cong A[G]$ as rings, and such that if $B$ is a ring and $H$ is a group, then $R\cong B[H]$ as rings if and only if there is a finite abelian group $J$ such that $B\cong A[J]$ as rings and $J\times H\cong G$ 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 $G$.