Results 141 to 150 of about 60,182 (167)
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Communications in Algebra, 2010
An element of a ring is unique clean if it can be uniquely written as the sum of an idempotent and a unit. A ring $R$ is uniquely $ $-clean if some power of every element in $R$ is uniquely clean. In this article, we prove that a ring $R$ is uniquely $ $-clean if and only if for any $a\in R$, there exists an $m\in {\Bbb N}$ and a central idempotent ...
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An element of a ring is unique clean if it can be uniquely written as the sum of an idempotent and a unit. A ring $R$ is uniquely $ $-clean if some power of every element in $R$ is uniquely clean. In this article, we prove that a ring $R$ is uniquely $ $-clean if and only if for any $a\in R$, there exists an $m\in {\Bbb N}$ and a central idempotent ...
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Communications in Algebra, 2016
A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A1 ⊕ A2 with M′ ≅ M, there is a decomposition M′ =M1 ⊕ M2 such that A = M′ ⊕ [A1 ∩ (M1 ⊕ B)] ⊕ [A2 ∩ (M2 ⊕ B)].
Hongbo Zhang, Victor Camillo
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A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A1 ⊕ A2 with M′ ≅ M, there is a decomposition M′ =M1 ⊕ M2 such that A = M′ ⊕ [A1 ∩ (M1 ⊕ B)] ⊕ [A2 ∩ (M2 ⊕ B)].
Hongbo Zhang, Victor Camillo
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Mediterranean Journal of Mathematics, 2019
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Marjan Sheibani Abdolyousefi +2 more
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Marjan Sheibani Abdolyousefi +2 more
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Summary: In this paper we introduce the concepts of special regular clean elements and regular clean decomposition in a ring \(R\). These concepts lead us to the notion of special regular clean ring. We prove that for a special regular clean element \(a = e + r\in R\) and unit \(u\in R\) then \(au\) is a special regular clean if \(u\) is an inner ...
Gogoi, Saurav, Saikia, Helen
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Gogoi, Saurav, Saikia, Helen
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Algebra Colloquium, 2007
Let R be a ring and g(x) a polynomial in C[x], where C=C(R) denotes the center of R. Camillo and Simón called the ring g(x)-clean if every element of R can be written as the sum of a unit and a root of g(x). In this paper, we prove that for a, b ∈ C, the ring R is clean and b-a is invertible in R if and only if R is g1(x)-clean, where g1(x)=(x-a)(x-b).
Wang, Zhou, Chen, Jianlong
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Let R be a ring and g(x) a polynomial in C[x], where C=C(R) denotes the center of R. Camillo and Simón called the ring g(x)-clean if every element of R can be written as the sum of a unit and a root of g(x). In this paper, we prove that for a, b ∈ C, the ring R is clean and b-a is invertible in R if and only if R is g1(x)-clean, where g1(x)=(x-a)(x-b).
Wang, Zhou, Chen, Jianlong
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Asian-European Journal of Mathematics
This study explores in depth the structure and properties of the so-called strongly[Formula: see text]-clean rings, that is a novel class of rings in which each ring element decomposes into a sum of a commuting idempotent and an element from the subset [Formula: see text] (see also [P. H. Tin and N. Q.
Peter Danchev +3 more
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This study explores in depth the structure and properties of the so-called strongly[Formula: see text]-clean rings, that is a novel class of rings in which each ring element decomposes into a sum of a commuting idempotent and an element from the subset [Formula: see text] (see also [P. H. Tin and N. Q.
Peter Danchev +3 more
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COMMUTATIVE INVO-CLEAN GROUP RINGS
Universal Journal of Mathematics and Mathematical Sciences, 2018Summary: A ring \(R\) is called invo-clean if any of its elements is the sum of an involution and an idempotent. For each ring \(R\) and each group \(G\), we find a criterion when the commutative group ring \(R[G]\) is invo-clean only in terms of \(R\), \(G\) and their sections. Our result is parallel to that of \textit{W. Wm. McGovern} [Int. J.
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Catalytic Enantioselective Ring-Opening Reactions of Cyclopropanes
Chemical Reviews, 2021Vincent Pirenne +2 more
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C–C Bond Cleavages of Cyclopropenes: Operating for Selective Ring-Opening Reactions
Chemical Reviews, 2021Ruben Vicente
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