Unit sum numbers of rings and modules

The additive unit structure of complex biquadratic fields

“…and α i + α j = 0 for i < j and (i, j) = (1, 2), (3,4). One sees at once that the cyclic permutation (1, 2, 3) of the indices of α does not induce an automorphisms.…”

“…Zelinsky's work gave rise to many investigations of rings that are generated by their units (see [9] for an overview). These investigations led Goldsmith, Pabst and Scott [4] to the following definition: Definition 1. Let R be a ring (with identity).…”

A Survey of Rings Generated by Units

“…Goldsmith, Pabst and Scott introduced the idea of studying rings whose elements can be written as the sum of a fixed number of units [8]. A ring R is said to have the n-sum property, for a positive integer n, if its every element can be written as the sum of exactly n units of R. The unit sum number of a ring R, denoted by usn(R), is the least integer n, if any such integer exists, such that R has the n-sum property.…”

“…In fact, each element of End D (V ) is the sum of two units except for one obvious case when V is a one-dimensional space over Z 2 . 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].…”

“…In fact, each element of End D (V ) is the sum of two units except for one obvious case when V is a one-dimensional space over Z 2 . 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.…”