Rings with unipotent units

Danchev, Peter V.; Lam, T. Y.

doi:10.5486/pmd.2016.7405

Cited by 53 publications

(49 citation statements)

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“…Thus, UU rings can be viewed as the "strong version" of UNC rings. The next lemma was proved for a nil-clean ring in [8] and for a UU ring in [7]. The next result is basic for studying the structure of a UNC ring.…”

Section: Introductionmentioning

confidence: 91%

“…These rings have been extensively investigated in Danchev and Lam [7], where, among others, it is proved that a ring is strongly nil-clean if and only if it is an exchange (clean) UU ring, and it is asked whether a clean ring R is nil-clean if and only if every unit of R is Proposition 2.2. [8] A unit u of R is strongly nil-clean if and only if u ∈ 1 + Nil(R).…”

Section: Introductionmentioning

confidence: 99%

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Rings in which every unit is a sum of a nilpotent and an idempotent

Karimi-Mansoub¹,

Koşan²,

Zhou³

2018

Advances in Rings and Modules

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A ring R is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in Cȃlugȃreanu [3] and in Danchev and Lam [7]. In this paper, two generalizations of UU rings are discussed. We study rings for which every unit is a sum of a nilpotent and an idempotent, and rings for which every unit is a sum of a nilpotent and two idempotents that commute with one another.1 2 KARIMI-MANSOUB, KOSAN, ZHOU nil-clean. Here we are motivated to study rings whose units are nil-clean. These rings will be called UNC rings.In section 2, we first prove several basic properties of UNC rings. Especially it is proved that every semilocal UNC ring is nil-clean. This can be seen as a partial answer to the question of Danchev and Lam. We next show that the matrix ring over a commutative ring R is a UNC ring if and only if R/J(R) is Boolean with J(R) nil. As a consequence, the matrix ring over a UNC ring need not be a UNC ring. We also discuss when a group ring is a UNC (UU) ring. In the last part of this section, it is shown that UU rings are exactly those rings whose units are uniquely nil-clean. As the main result in this section, it is proved that a ring R is strongly nil-clean if and only if R is a semipotent UNC ring.As another natural generalization of UU rings, in section 3 we determine the rings for which every unit is a sum of a nilpotent and two idempotents that commute with one another. We also deal with a special case where every unit of the ring is a sum of two commuting idempotents. These conditions can be compared with the so-called strongly 2nil-clean rings introduced by Chen and Sheibani [4], and the rings for which every element is a sum of two commuting idempotents, studied by Hirano and Tominaga in [11].We write M n (R), T n (R) and R[t] for the n × n matrix ring, the n × n upper triangular matrix ring, and the polynomial ring over R, respectively. For an endomorphism σ of a ring R, let R[t; σ] denote the ring of left skew power series over R. Thus, elements of R[t; σ] are polynomials in t with coefficients in R written on the left, subject to the relation tr = σ(r)t for all r ∈ R. The group ring of a group G over a ring R is denoted by RG. Units being nil-clean2.1. Basic properties. We present various properties of the rings whose units are nil-clean, and prove that every semilocal ring whose units are nil-clean is a nil-clean ring.Definition 2.1. A ring R is called a UNC ring if every unit of R is nil-clean.Every nil-clean ring is a UNC ring. A ring R is called a UU ring if U (R) = 1 + Nil(R) (see [3]).

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Rings in which every unit is a sum of a nilpotent and an idempotent

Karimi-Mansoub¹,

Koşan²,

Zhou³

2018

Advances in Rings and Modules

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“…Proof. [6,Theorem 3.2. ] states that if R is a ring satisfying assumptions of the lemma, then (1 − r) m s = 1, for any r ∈ J(R).…”

Section: This Implies Thatmentioning

confidence: 99%

𝑛-torsion clean rings

Danchev¹,

Matczuk²

2019

Contemporary Mathematics

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“…Thus we are now ready to proceed by proving with the following. Enlarging now the main concept of UU rings from [5] as those rings R such that U (R) = 1 + N il(R), we define R to be a ring with unipotent involutions or briefly a UI ring, provided Inv(R) ⊆ 1 + N il(R) (see [4,Problem 6]). Surprisingly, this condition does not give nothing new.…”

Section: Invo-clean Ringsmentioning

confidence: 99%

“…In [5] it was proved that clean UU rings are strongly nil-clean. It is thereby reasonably adequate to conjecture that clean UI rings are nil-clean, but in view of Proposition 2.20 together with some standard facts this will be completely wrong.…”

Section: Invo-clean Ringsmentioning

confidence: 99%