Two Classes of Rings Generated by Their Units

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Mathematics (HMC)

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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 2n2 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 Rn is a sum of two units. A variety of conditions are produced that are either necessary or sufficient for every element of Rn to be a sum of two units if n > 1.

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© 1974 Elsevier