#### Title

Two Classes of Rings Generated by Their Units

#### Document Type

Article

#### Department

Mathematics (HMC)

#### Publication Date

7-1974

#### Abstract

In 1953 and 1954, K. Wolfson and D. Zelinsky showed, independently, that every element of the ring of all linear transformations of a vector space over a division ring of characteristic not 2 is a sum of two nonsingular ones, see [16] and [17]. In 1958, Skornyakov [15, p. 1671 posed the problem of determining which regular rings are generated by their units. In 1969, while apparently unaware of Skornyakov’s book, G. Ehrlich [3] produced a large class of regular rings generated by their units; namely, those rings *R* with identity in which 2 is a unit and are such that for every *a ε R* there is a unit *u ε R* such that *aua = a*. (See also [9] where this author obtained other characterizations of such regular rings.) Finally, in [14], *R*. Raphael launched a systematic study of rings generated by their units, which he calls *S*-rings.

This note is devoted mainly to generalizing two theorems of Raphael. He shows in [14] that if *R* is any ring with identity, and *n* > 1 is a positive integer, then every element of the ring *R*, of all *n* x *n* matrices with entries from *R* is a sum of *2n ^{2}* units. In Section 1 I show, under the same assumptions, that every element of R, is a sum of three units, and I produce a class of rings

*R*such that not every element of

*R*is a sum of two units. A variety of conditions are produced that are either necessary or sufficient for every element of

_{n}*R*to be a sum of two units if

_{n}*n*> 1.

#### Rights Information

© 1974 Elsevier

#### DOI

10.1016/0021-8693(74)90013-1

#### Recommended Citation

Henriksen, Melvin. 1974. Two classes of rings generated by their units. Journal of Algebra. 31(1):182-193.