Computing the unit group of a commutative finite $\mathbb{Z}$-algebra

TL;DR

This paper presents algorithms for computing the structure and generators of the unit group in commutative finite $Z$-algebras, with implementations in SageMath and detailed complexity analysis.

Contribution

It introduces new algorithms for explicitly computing the unit group structure in finite $Z$-algebras, including reductions to simpler cases and implementation details.

Findings

Algorithms successfully compute unit groups for various cases.

Implementation available as a SageMath package.

Complexity analysis provided for the algorithms.

Abstract

For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to compute generators and the structure of this group. This is achieved by reducing the task first to the case of reduced rings, then to torsion-free reduced rings, and finally to an order in a reduced ring. The simplified cases are treated via a calculation of exponent lattices and various algorithms to compute the minimal primes, primitive idempotents, and other basic objects. All algorithms have been implemented and are available as a SageMath package. Whenever possible, the time complexity of the described methods is tracked carefully.