A Survey of Rings Generated by Units

Abstract

— This article presents a brief survey of the work done on rings generated by their units. RESUME. — Cet article est un bref survey de l’etude des anneaux engendres par leurs unites. The study of rings generated additively by their units started in 19531954 when K. G. Wolfson [24] and D. Zelinsky [25] proved, independently, that every linear transformation of a vector space V over a division ring D is the sum of two nonsingular linear transformations, except when dim V = 1 and D = Z2 := Z/2Z. This implies that the ring of linear transformations EndD(V ) is generated additively by its unit elements. In fact, each element of EndD(V ) is the sum of two units except for one obvious case when V is a one-dimensional space over Z2. In 1998 this result was reproved by Goldsmith, Pabst and Scott, who remarked that this result can hardly be new, but they were unable to find any reference to it in the literature [8]. Wolfson and Zelinsky’s result generated quite a bit of interest in the study of rings that are generated by their unit elements. All our rings are associative (not necessarily commutative) with identity element 1. A ring R is called a von Neumann regular ring if for each element x ∈ R there exists an element y ∈ R such that xyx = x. In 1958 Skornyakov ([20], Problem 31, page 167) asked: Is every element of a von Neumann regular ring (which does not have Z2 as a factor ring) a sum of units? This