Additive unit representations in rings over global fields -- A survey

Abstract

We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. We focus on rings of integers in number fields and in function fields of one variable over perfect fields. The central problem is whether and how certain rings are (additively) generated by their units. In the final section we deal with matrix rings over quaternions and over Dedekind domains. Our point of view is number-theoretic whereas we do not discuss the general algebraic background. 1. The unit sum number In 1954, Zelinsky [44] proved that every endomorphism of a vector space V over a division ring D is a sum of two automorphisms, except if D = Z/2Z and dimV = 1. This was motivated by investigations of Dieudonne on Galois theory of simple and semisimple rings [7] and was probably the first result about the additive unit structure of a ring. Using the terminology of Vamos [41], we say that an element r of a ring R (with unity 1, not necessarily commutative) is k-good if r is a sum of exactly k units of R. If every element of R has this property then we call R k-good. By Zelinsky’s result, the endomorphism ring of a vector space with more than two elements is 2-good. Clearly, if R is k-good then it is also l-good for every integer l > k. Indeed, we can write any element of R as r = (r − (l − k) · 1) + (l − k) · 1, and expressing r − (l − k) · 1 as a sum of k units gives a representation of r as a sum of l units. Goldsmith, Pabst and Scott [20] defined the unit sum number u(R) of a ring R to be the minimal integer k such that R is k-good, if such an integer exists. If R is not k-good for any k then we put u(R) := ω if every element of R is a sum of units, and u(R) :=∞ if not. We use the convention k < ω <∞ for all integers k. Clearly, u(R) ≤ ω if and only if R is generated by its units. Here are some examples from [20] and [41]: • u(Z) = ω, 1991 Mathematics Subject Classification. 00-02, 11R27, 16U60.