Fine rings: A new class of simple rings

Călugăreanu, Grigore; Lam, T. Y.

doi:10.1142/s0219498816501735

Cited by 32 publications

(21 citation statements)

References 20 publications

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“…(ii) Invoking [8], every non-zero matrix is the sum of a unit and a nilpotent. Moreover, since F is a potent field, each invertible matrix is indeed a torsion unit, because as already showed above in point (i) it can be embedded into a finite field, so the result follows now directly.…”

Section: Some Main Results and Examplesmentioning

confidence: 99%

Rings in Which all Elements are Sums or Differences of Nilpotents and Potents

Abyzov¹,

Cohen²,

Danchev³

et al. 2021

Preprint

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We examine those rings in which the elements are sums or differences of nilpotents and potents (also including in some special cases tripotents). Such decompositions of matrices over certain rings and fields are also studied. These results of ours somewhat support recent achievements presented in a publication due to Abyzov-Tapkin (Siber. Math. J., 2021).In particular, letting n be an arbitrary natural number, the class of weakly n-torsion clean rings is defined and studied. As direct applications of the facts presented above, some characterization theorems describing the structure of these rings up to an isomorphism and their matrix analogs, are established.The obtained results of ours somewhat supply recent achievements presented in a publication due to Danchev-Matczuk (Contemp. Math., 2019).2010 Mathematics Subject Classification. 16D60; 16S34; 16U60. Key words and phrases. clean rings, strongly clean rings, n-torsion clean rings, weakly ntorsion clean rings, idempotents, tripotents, potents, nilpotents, units.

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Section: Some Main Results and Examplesmentioning

confidence: 99%

Rings in Which all Elements are Sums or Differences of Nilpotents and Potents

Abyzov¹,

Cohen²,

Danchev³

et al. 2021

Preprint

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“…We start by recalling three special types of elements in a ring. An element in a ring is 2-good if it is a sum of two units (see [19]), is fine if it a sum of a unit and a nilpotent (see [6]), and is 2-nilgood 1 if it is a sum of two nilpotents (see [5]). By the terminology of Vámos [19], a ring is 2-good if each of its elements is 2-good.…”

Section: Introductionmentioning

confidence: 99%

When is every non central-unit a sum of two nilpotents?

Breaz¹,

Zhou²

2021

Preprint

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Motivated by the notions of 2-good rings and fine rings, we define a 2-nilgood ring as a ring whose every non central-unit is the sum of two nilpotents. This is a study of 2-nilgood rings. The following results are obtained: (a) a ring is 2-nilgood iff it is either a 2-nilgood simple ring or a commutative local ring with nil Jacobson radical; (b) a 2-nilgood simple ring is a field if it is right Goldie; (c) a 2-nilgood ring containing a uniform right ideal is a commutative local ring with nil Jacobson radical; (d) a ring whose every non central-unit is a sum of a square-zero element and a nilpotent is a commutative local ring with nil Jacobson radical. We conclude with the conjecture that a 2-nilgood simple ring is a field.

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“…In this paper we will not consider the other natural generalization to non-commutative rings; in fact, all algebraic structures will be commutative throughout the paper. Nevertheless, let us mention at least one of the latest results on simple non-commutative rings [6]. Definition 1.1.…”

Section: Introductionmentioning

confidence: 99%

Idempotence of finitely generated commutative semifields

Kala

Korbelář

2018

Forum Mathematicum

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We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.2010 Mathematics Subject Classification. primary: 12K10, 16Y60, 20M14, secondary: 11H06, 52A20.This statement can be now understood as an extension of the folklore theorem referred in the beginning, i.e., that every (commutative) field that is finitely generated as a ring is finite. Of course, the greatest difference is the existence of additively idempotent semifields.Since every semifield is clearly ideal-simple, part (b) of the theorem follows immediately from (c). Also, one can be slightly more precise in the additively constant case: this occurs if and only if there is a finitely generated (multiplicative) abelian group G(·) and the semiring is the semifield S := G ∪ {o}, where o is a new element. Operations that extend the multiplication on G are defined by a+b = o and a·o = o for all a, b ∈ S.

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Fine rings: A new class of simple rings

Cited by 32 publications

References 20 publications

Rings in Which all Elements are Sums or Differences of Nilpotents and Potents

Rings in Which all Elements are Sums or Differences of Nilpotents and Potents

When is every non central-unit a sum of two nilpotents?

Idempotence of finitely generated commutative semifields

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