On the unit sum number of some rings

Ashrafi, Nahid

doi:10.1093/qmath/hah023

Cited by 36 publications

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“…Moreover, Belcher [2] characterized all quadratic fields whose rings of integers are generated by their units. About thirty years later Ashrafi and Vámos [1], Jarden and Narkiewicz [6] and Tichy and Ziegler [8] resumed this topic. In particular Ashrafi and Vámos showed that the ring of integers of quadratic fields, complex cubic fields and fields of the form Q(ζ 2 n ), with ζ 2 n is a 2 n -th primitive root of unity, do not have finite unit sum number.…”

Section: Introductionmentioning

confidence: 99%

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The additive unit structure of complex biquadratic fields

Ziegler¹

2008

Glas. Mat. Ser. III

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Section: Introductionmentioning

confidence: 99%

“…Tichy and Ziegler characterized all purely cubic number fields whose ring of integers are generated by their units. Note that the case of quadratic fields has been rediscovered by Ashrafi and Vámos [1].…”

Section: Introductionmentioning

confidence: 99%

The additive unit structure of complex biquadratic fields

Ziegler¹

2008

Glas. Mat. Ser. III

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“…If |D| ≥ 3, then u(D) = 2; whereas if |D| = 2, that is, D = ‫ޚ‬ 2 , the field of two elements, then u(‫ޚ‬ 2 ) = ω. We have also u(‫ޚ‬ 2 × ‫ޚ‬ 2 ) = ∞ -see [Ashrafi and Vámos 2005] for unit sum numbers of some other rings. The topic of unit sum numbers seems to have arisen with a paper by Zelinsky [1954], in which he shows that if V is any finite-or infinite-dimensional vector space over a division ring D, then every linear transformation is the sum of two automorphisms unless dim V = 1 and D is the field of two elements.…”

Section: Preliminaries and The Main Resultsmentioning

confidence: 98%

“…Interest in this topic increased recently after Goldsmith, Pabst and Scott [1998] defined the unit sum number. For additional historical background, see [Vámos 2005], which also contains references to recent work in this area.…”

Section: Preliminaries and The Main Resultsmentioning

confidence: 99%

Necessary and sufficient conditions for unit graphs to be Hamiltonian

Maimani¹,

Pournaki²,

Yassemi³

2011

Pacific J. Math.

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“…Ashrafi and Vámos studied the unit sum numbers of rings of integers of number fields [1]. Let K = Q(ξ) be a number field (that is, a finite extension of Q) and let O K be the ring of integers of K. For details on the ring of integers of a number field, the reader is referred to [14].…”

Section: Henriksen's Question (Question E Page 192 [11])mentioning

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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On the unit sum number of some rings

Cited by 36 publications

References 0 publications

The additive unit structure of complex biquadratic fields

The additive unit structure of complex biquadratic fields

Necessary and sufficient conditions for unit graphs to be Hamiltonian

A Survey of Rings Generated by Units

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