Determining unit groups and $\mathrm{K}_1$ of finite rings
Tommy Hofmann
TL;DR
The paper studies the problem of computing an effective presentation of the unit group $R^{\times}$ of a finite ring $R$, together with a way to express elements as words in generators, and shows that this task is probabilistic polynomial-time equivalent to classical number-theoretic problems such as factoring and discrete logarithms in finite fields. It develops an extension-patching framework that decomposes $R^{\times}$ into a unipotent part $1+J$ and a semisimple quotient $(R/J)^{\times}$ and then combines presentations for these components, enabling full computation for arbitrary finite rings. The authors provide concrete algorithms for unipotent units, the semisimple case, and the general case, and extend the results to abelianizations $R^{\times\mathrm{ab}}$ and the first $K$-group $K_1(R)$, with reductions across related problems. They give broad applications to finite-dimensional algebras over finite fields, finite quotient rings of arithmetic orders, and group rings, together with implementation and numerical examples that demonstrate practical performance and applicability to noncommutative arithmetic orders. Overall, the work links unit-group structure to fundamental number-theoretic challenges, enabling practical computation in noncommutative algebra settings relevant to arithmetic and group-ring problems.
Abstract
We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem is equivalent to the number theoretic problems of factoring integers and solving discrete logarithms in finite fields. A similar equivalence is shown for the problem of determining the abelianization of the unit group or the first $K$-group of finite rings.