Rings generated by their units

Fisher, Joe W.; Snider, Robert L.

doi:10.1016/0021-8693(76)90103-4

Cited by 40 publications

(21 citation statements)

References 8 publications

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“…If this supremum is finite, then I is said to have bounded index of nilpotence. Fisher and Snider [6] showed that a von Neumann regular ring with primitive factor rings artinian is unit-regular. Therefore, in view of the result of Ehrlich, they obtained the following.…”

Section: Von Neumann Regular Rings Generated By Unitsmentioning

confidence: 99%

A Survey of Rings Generated by Units

Srivastava¹

2010

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Abstract. This article presents a brief survey of the work done on rings generated by their units.The study of rings generated additively by their units started in 1953-1954 and D. Zelinsky [25] proved, independently, that every linear transformation of a vector space V over a division ring D is the sum of two nonsingular linear transformations, except when dim V = 1 and D = Z 2 := Z/2Z. This implies that the ring of linear transformations End D (V ) is generated additively by its unit elements. 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. In 1958 Skornyakov ([20], Problem 31, page 167) asked: Is every element of a von Neumann regular ring (which does not have Z 2 as a factor ring) a sum of units? This question of Skornyakov was answered in the negative by Bergman in 1977 (see [9]). Bergman constructed a von Neumann regular algebra in which not all elements are sums of units.There are several natural classes of rings that are generated by their unit elements -in particular, rings in which each element is the sum of two units. Let X be a completely regular Hausdorff space. Then every element in the ring of real-valued continuous functions on X is the sum of two units [18]. Every element in a real or complex Banach algebra is the sum of two units [22].The rings in which each element is the sum of k units were called (S, k)-rings by Henriksen. Vámos has called such rings k-good rings.It may be easily observed that if R is a 2-good ring, then the matrix ring M n (R) is also a 2-good ring. The following result of Henriksen is quite surprising, and it shows that every ring is Morita equivalent to an (S, 3)-ring (or a 3-good ring). Theorem 1. (Henriksen, [11]) Let R be any ring. Then each element of the matrix ring M n (R), n > 1, can be written as the sum of exactly three units.We say that an n × n matrix A over a ring R admits a diagonal reduction if there exist invertible matrices P, Q ∈ M n (R) such that P AQ is a diagonal matrix.

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Section: Von Neumann Regular Rings Generated By Unitsmentioning

confidence: 99%

A Survey of Rings Generated by Units

Srivastava¹

2010

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…By Theorem 3.11, the prime ring R/P is finite. Thus, every prime factor ring of R is Artinian, so R is unit regular by Fisher-Snyder [5,Theorem 1.1].…”

Section: A Nonzero Regular Ring R Is a Division Ring Or A Finite Ringmentioning

confidence: 99%

Generalized inverses in regular rings

Savage¹

1980

Pacific J. Math.

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“…In 1974, Raphael [7] launched a systematic study of rings generated by their units, which he calls S-rings. Finally, in 1976, Fisher and Snider [3] proved that if R is a von Neumann regular ring with primitive factor rings artinian and 2 is a unit, then every element of R can be expressed as the sum of two units. In 1998, Wolfson's and Zelinsky's result was reproved by Goldsmith, Pabst and Scott where they remarked that this result can hardly be new but they were unable to find any reference to it in the literature (see [4]).…”

Section: Introductionmentioning

confidence: 99%