A survey of rings generated by units.
The research topic surveyed arose from the result of K. G. Wolfson and D. Zelinsky from the 50's that every linear transformation of a vector space over a division ring but the two-element space is the sum of two units. Many surprising results have been found since then, such as construction of a von Neumann regular algebra in which not all elements are sums of units; and that the matrix algebra of at least \(2\times2\) matrices over any ring has unit sum number 3, that is, all elements are sums of exactly 3 units, due to \textit{M. Henriksen } [J. Algebra 31, 182-193 (1974; Zbl 0285.16009)]. The following research areas are described in detail: von Neumann regular rings generated additively by units; right self-injective rings generated additively by units; endomorphism rings of variously conditioned modules, such as (quasi) injective modules, generated additively by units; unit sum numbers of variously conditioned rings such as right self-injective rings and rings of integers. Furthermore, generalization of the problem with regular elements instead of units (which coincide in von Neumann regular rings) is also surveyed, and several open problems are propounded.
scientific article; zbMATH DE number 5799087
scientific article