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The Group Structure of Unit Г-regular Ring Elements
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Let be a prime-ring with 2-torsion free, a nonzero ideal of M and : → a left generalized derivation of , with associated nonzero derivation d on. If () ∈ () for all ∈ , then is a commutative-ring.
The existence of the Moore-Penrose inverse is discussed for elements of a *-regular ring R. A technique is developed for computing conditional and reflexive inverses for matrices in R22, which is then used to calculate the Moore-Penrose inverse for these matrices. Several applications are given, generalizing many of the classical results; in particular, we shall emphasize the cases of bordered matrices, Schur complements, block-rank formulae and EP elements.
Local properties of unit regular ring elements are investigated. It is shown that an element of a ring R with unity is regular if and only if there exists a unit u ϵ R and a group G such that a ϵ uG. © 1977, University of California, Berkeley. All Rights Reserved.
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.
- M P Drazin
M.P.Drazin, Pseudo-inverse in associative rings and semigroups, American
Mathematics Monthly, 65 (1958) 506-551.
- G Ehrlich
G. Ehrlich, Units regular rings, Portugal Mathematics, 27 (1969)209-212.