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Rings Whose Clean Elements are Uniquely Clean
Mediterranean Journal of Mathematics, 2022\textit{W. K. Nicholson} defined in [Trans. Am. Math. Soc. 229, 269--278 (1977; Zbl 0352.16006)] a ring \(R\) to be \textit{clean} if each its element \(r\) is \textit{clean}, that is, \(r=u+e\), where \(u\in R\) is an invertible element and \(e\in R\) is an idempotent element.
Grigore Călugăreanu, Yiqiang Zhou
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Journal of Algebra and Its Applications, 2014
A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and ...
Li, Yuanlin +2 more
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A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and ...
Li, Yuanlin +2 more
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Iranian Journal of Science and Technology, Transactions A: Science, 2017
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Ashrafi, Nahid, Nasibi, Ebrahim
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Ashrafi, Nahid, Nasibi, Ebrahim
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Vietnam Journal of Mathematics, 2016
Let \(R\) be a ring with identity. An element \(x\in R\) is called (von Neumann) regular if there exists \(y\in R\) such that \(x=xyx\). An element \(a\in R\) is called \(NR\)-clean if \(a=x+b\), where \(x\) is a regular element in \(R\) and \(b\) is a nilpotent element in \(R\). When every element of \(R\) is \(NR\)-clean, \(R\) is called \(NR\)-clean
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Let \(R\) be a ring with identity. An element \(x\in R\) is called (von Neumann) regular if there exists \(y\in R\) such that \(x=xyx\). An element \(a\in R\) is called \(NR\)-clean if \(a=x+b\), where \(x\) is a regular element in \(R\) and \(b\) is a nilpotent element in \(R\). When every element of \(R\) is \(NR\)-clean, \(R\) is called \(NR\)-clean
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Algebra Colloquium, 2015
A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG.
Gao, Yanyan, Chen, Jianlong, Li, Yuanlin
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A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG.
Gao, Yanyan, Chen, Jianlong, Li, Yuanlin
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Czechoslovak Mathematical Journal, 2023
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Sheibani Abdolyousefi, Marjan +1 more
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Sheibani Abdolyousefi, Marjan +1 more
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Journal of Algebra and Its Applications, 2011
A *-ring R is called a *-clean ring if every element of R is the sum of a unit and a projection, and R is called a strongly *-clean ring if every element of R is the sum of a unit and a projection that commute with each other. These concepts were introduced and discussed recently by [L.
Li, Chunna, Zhou, Yiqiang
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A *-ring R is called a *-clean ring if every element of R is the sum of a unit and a projection, and R is called a strongly *-clean ring if every element of R is the sum of a unit and a projection that commute with each other. These concepts were introduced and discussed recently by [L.
Li, Chunna, Zhou, Yiqiang
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Mediterranean Journal of Mathematics, 2019
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Chen, Huanyin +2 more
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Chen, Huanyin +2 more
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Nil-clean and strongly nil-clean rings
Journal of Pure and Applied Algebra, 2016Let \(a\) be an element of a ring \(R\) with identity. Then \(a\) is called \textit{clean} if \(a\) can be written as \(a=e+b\) where \(e=e^2\in R\) is an idempotent and \(b\in R\) is a unit. If this can be done in such a way that \(eb=be\), then \(a\) is called \textit{strongly clean}.
Zhou, Yiqiang +2 more
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CLEAN, ALMOST CLEAN, POTENT COMMUTATIVE RINGS
Journal of Algebra and Its Applications, 2007We give a complete characterization of the class of commutative rings R possessing the property that Spec(R) is weakly 0-dimensional. They turn out to be the same as strongly π-regular rings. We considerably strengthen the results of K. Samei [13] tying up cleanness of R with the zero dimensionality of Max(R) in the Zariski topology.
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