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Yiqiang Zhou - One of the best experts on this subject based on the ideXlab platform.
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Nil-clean and strongly nil-clean Rings
Journal of Pure and Applied Algebra, 2016Co-Authors: M. Tamer Koşan, Zhou Wang, Yiqiang ZhouAbstract:An element a of a Ring R is nil-clean if a=e+b where e2=e∈R and b is a nilpotent; if further eb=be, the element a is called strongly nil-clean. The Ring R is called nil-clean (resp., strongly nil-clean) if each of its elements is nil-clean (resp., strongly nil-clean). It is proved that an element a is strongly nil-clean iff a is a sum of an idempotent and a unit that commute and a−a2 is a nilpotent, and that a Ring R is strongly nil-clean iff R/J(R) is boolean and J(R) is nil, where J(R) denotes the Jacobson radical of R. The strong nil-cleanness of Morita contexts, formal Matrix Rings and group Rings is discussed in details. A necessary and sufficient condition is obtained for an ideal I of R to have the property that R/I strongly nil-clean implies R is strongly nil-clean. Finally, responding to the question of when a Matrix Ring is nil-clean, we prove that the Matrix Ring over a 2-primal Ring R is nil-clean iff R/J(R) is boolean and J(R) is nil, i.e., R is strongly nil-clean.
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when is every Matrix over a division Ring a sum of an idempotent and a nilpotent
Linear Algebra and its Applications, 2014Co-Authors: Tamer M Kosan, Tsiukwen Lee, Yiqiang ZhouAbstract:Abstract A Ring is called nil-clean if each of its elements is a sum of an idempotent and a nilpotent. In response to a question of S. Breaz et al. in [1] , we prove that the n × n Matrix Ring over a division Ring D is a nil-clean Ring if and only if D ≅ F 2 . As consequences, it is shown that the n × n Matrix Ring over a strongly regular Ring R is a nil-clean Ring if and only if R is a Boolean Ring, and that a semilocal Ring R is nil-clean if and only if its Jacobson radical J ( R ) is nil and R / J ( R ) is a direct product of Matrix Rings over F 2 .
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A class of formal Matrix Rings
Linear Algebra and its Applications, 2013Co-Authors: Gaohua Tang, Yiqiang ZhouAbstract:Abstract This paper concerns the formal Matrix Ring M n ( R ; s ) over a Ring R defined by a central element s in R . Various basic properties of these Rings are established; the isomorphism problem between these Rings is addressed; some known results on the usual Matrix Rings are extended to the formal Matrix Rings; and the Cayley–Hamilton Theorem is proved for matrices in a formal Matrix Ring.
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strong cleanness of the 2 2 Matrix Ring over a general local Ring
Journal of Algebra, 2008Co-Authors: Xiande Yang, Yiqiang ZhouAbstract:Abstract A Ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [G. Borooah, A.J. Diesl, T.J. Dorsey, Strongly clean Matrix Rings over commutative local Rings, J. Pure Appl. Algebra 212 (1) (2008) 281–296] completely characterized the commutative local Rings R for which M n ( R ) is strongly clean. For a general local Ring R and n > 1 , however, it is unknown when the Matrix Ring M n ( R ) is strongly clean. Here we completely characterize the local Rings R for which M 2 ( R ) is strongly clean.
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strong cleanness of the 2 times 2 Matrix Ring over a general local Ring
arXiv: Rings and Algebras, 2008Co-Authors: Xiande Yang, Yiqiang ZhouAbstract:A Ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the commutative local Rings $R$ for which ${\mathbb M}_n(R)$ is strongly clean. For a general local Ring $R$ and $n>1$, however, it is unknown when the Matrix Ring ${\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local Rings $R$ for which ${\mathbb M}_2(R)$ is strongly clean.
Xiande Yang - One of the best experts on this subject based on the ideXlab platform.
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A Note on Strongly Clean Matrix Rings
Communications in Algebra, 2010Co-Authors: Lingling Fan, Xiande YangAbstract:Let R be an associative Ring with identity. An element a ∈ R is called strongly clean if a = e + u with e 2 = e ∈ R, u a unit of R, and eu = ue. A Ring R is called strongly clean if every element of R is strongly clean. Strongly clean Rings were introduced by Nicholson [7]. It is unknown yet when a Matrix Ring over a strongly clean Ring is strongly clean. Several articles discussed this topic when R is local or strongly π-regular. In this note, necessary conditions for the Matrix Ring 𝕄 n (R) (n > 1) over an arbitrary Ring R to be strongly clean are given, and the strongly clean property of 𝕄2(RC 2) over the group Ring RC 2 with R local is obtained.
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strong cleanness of the 2 2 Matrix Ring over a general local Ring
Journal of Algebra, 2008Co-Authors: Xiande Yang, Yiqiang ZhouAbstract:Abstract A Ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [G. Borooah, A.J. Diesl, T.J. Dorsey, Strongly clean Matrix Rings over commutative local Rings, J. Pure Appl. Algebra 212 (1) (2008) 281–296] completely characterized the commutative local Rings R for which M n ( R ) is strongly clean. For a general local Ring R and n > 1 , however, it is unknown when the Matrix Ring M n ( R ) is strongly clean. Here we completely characterize the local Rings R for which M 2 ( R ) is strongly clean.
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strong cleanness of the 2 times 2 Matrix Ring over a general local Ring
arXiv: Rings and Algebras, 2008Co-Authors: Xiande Yang, Yiqiang ZhouAbstract:A Ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the commutative local Rings $R$ for which ${\mathbb M}_n(R)$ is strongly clean. For a general local Ring $R$ and $n>1$, however, it is unknown when the Matrix Ring ${\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local Rings $R$ for which ${\mathbb M}_2(R)$ is strongly clean.
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Strongly clean Rings and g(x)-clean Rings
2007Co-Authors: Xiande YangAbstract:Let R be an associative Ring with identity 1 ≠ 0. An element a ∈ R is called clean if there exists an idempotent e and a unit u in R such that a = e + u, and a is called strongly clean if, in addition, eu = ue. The Ring R is called clean (resp., strongly clean) if every element of R is clean (resp., strongly clean). The notion of a clean Ring was given by Nicholson in 1977 in a study of exchange Rings and that of a strongly clean Ring was introduced also by Nicholson in 1999 as a natural generalization of strongly π-regular Rings. Besides strongly π-regular Rings, local Rings give another family of strongly clean Rings. The main part of this thesis deals with the question of when a Matrix Ring is strongly clean. This is motivated by a counter-example discovered by Sanchez Campos and Wang-Chen respectively to a question of Nicholson whether a Matrix Ring over a strongly clean Ring is again strongly clean. They both proved that the 2 x 2 Matrix Ring M₂(Z₍₂₎) is not strongly clean, where Z₍₂₎ is the localization of Z at the prime ideal (2). The following results are obtained regarding this question: • Various examples of non-strongly clean Matrix Rings over strongly clean Rings. • Completely determining the local Rings R (commutative or noncommutative) for which M₂ (R) is strongly clean. • A necessary condition for M₂(R) over an arbitrary Ring R to be strongly clean. • A criterion for a single Matrix in Mn(R) to be strongly clean when R has IBN and every finitely generated projective R-module is free. • A sufficient condition for the Matrix Ring Mn(R) over a commutative Ring R to be strongly clean. • Necessary and sufficient conditions for Mn(R) over a commutative local Ring R to be strongly clean. • A family of strongly clean triangular Matrix Rings. • New families of strongly π-regular (of course strongly clean) Matrix Rings over noncommutative local Rings or strongly π-regular Rings. Another part of this thesis is about the so-called g(x)-clean Rings. Let C(R) be the center of R and let g(x) be a polynomial in C(R)[x]. An element a ∈ R is called g(x)clean if a == e + u where g(e) == 0 and u is a unit of R. The Ring R is g(x)-clean if every element of R is g(x)-clean. The (x² - x )-clean Rings are precisely the clean Rings. The notation of a g(x)-clean Ring was introduced by Camillo and Simon in 2002. The relationship between clean Rings and g(x)-clean Rings is discussed here.
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on strongly clean Matrix and triangular Matrix Rings
Communications in Algebra, 2006Co-Authors: Jianlong Chen, Xiande Yang, Yiqiang ZhouAbstract:A Ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean Rings are “additive analogs” of strongly regular Rings, where a Ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean Rings were introduced in Nicholson (1999) where their connection with strongly π-regular Rings and hence to Fitting's Lemma were discussed. Local Rings and strongly π-regular Rings are all strongly clean. In this article, we identify new families of strongly clean Rings through Matrix Rings and triangular Matrix Rings. For instance, it is proven that the 2 × 2 Matrix Ring over the Ring of p-adic integers and the triangular Matrix Ring over a commutative semiperfect Ring are all strongly clean.
Jianlong Chen - One of the best experts on this subject based on the ideXlab platform.
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when is a 2 2 Matrix Ring over a commutative local Ring quasipolar
Communications in Algebra, 2011Co-Authors: Jianlong ChenAbstract:A Ring R is quasipolar if for any a ∈ R, there exists p 2 = p ∈ R such that p ∈ comm2(a), p + a ∈ U(R) and ap ∈ R qnil . In this article, we determine when a 2 × 2 Matrix over a commutative local Ring is quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a Matrix to be quasipolar. Consequently, we obtain several equivalent conditions for the 2 × 2 Matrix Ring over a commutative local Ring to be quasipolar. Furthermore, it is shown that the 2 × 2 Matrix Ring over the Ring of p-adic integers is quasipolar.
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on strongly clean Matrix and triangular Matrix Rings
Communications in Algebra, 2006Co-Authors: Jianlong Chen, Xiande Yang, Yiqiang ZhouAbstract:A Ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean Rings are “additive analogs” of strongly regular Rings, where a Ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean Rings were introduced in Nicholson (1999) where their connection with strongly π-regular Rings and hence to Fitting's Lemma were discussed. Local Rings and strongly π-regular Rings are all strongly clean. In this article, we identify new families of strongly clean Rings through Matrix Rings and triangular Matrix Rings. For instance, it is proven that the 2 × 2 Matrix Ring over the Ring of p-adic integers and the triangular Matrix Ring over a commutative semiperfect Ring are all strongly clean.
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when is the 2 2 Matrix Ring over a commutative local Ring strongly clean
Journal of Algebra, 2006Co-Authors: Jianlong Chen, Xiande Yang, Yiqiang ZhouAbstract:A Ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. Local Rings are strongly clean. It is unknown when a Matrix Ring is strongly clean. However it is known from [J. Chen, X. Yang, Y. Zhou, On strongly clean Matrix and triangular Matrix Rings, preprint, 2005] that for any prime number p, the 2×2 Matrix Ring M2(Zˆp) is strongly clean where Zˆp is the Ring of p-adic integers, but M2(Z(p)) is not strongly clean where Z(p) is the localization of Z at the prime ideal generated by p. Let R be a commutative local Ring. A criterion in terms of solvability of a simple quadratic equation in R is obtained for M2(R) to be strongly clean. As consequences, M2(R) is strongly clean iff M2(R〚x〛) is strongly clean iff M2(R[x]/(xn)) is strongly clean iff M2(RC2) is strongly clean.
Leon Van Wyk - One of the best experts on this subject based on the ideXlab platform.
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Invertibility and Dedekind finiteness in structural Matrix Rings
Linear & Multilinear Algebra, 2011Co-Authors: Stephan Foldes, Jenő Szigeti, Leon Van WykAbstract:An example in Szigeti and van Wyk [J. Szigeti and L. van Wyk, SubRings which are closed with respect to taking the inverse, J. Algebra 318 (2007), pp. 1068–1076] suggests that Dedekind finiteness may play a crucial role in a characterization of the structural subRings M n (θ, R) of the full n × n Matrix Ring M n (R) over a Ring R, which are closed with respect to taking inverses. It turns out that M n (θ, R) is closed with respect to taking inverses in M n (R) if all the equivalence classes with respect to θ ∩ θ−1, except possibly one, are of a size less than or equal to p (say) and M p (R) is Dedekind finite. Another purpose of this article is to show that M n (θ, R) is Dedekind finite if and only if M m (R) is Dedekind finite, where m is the maximum size of the equivalence classes (with respect to θ ∩ θ−1). This provides a positive result for the inheritance of Dedekind finiteness by a Matrix Ring (albeit not a full Matrix Ring) from a smaller (full) Matrix Ring.
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subRings which are closed with respect to taking the inverse
Journal of Algebra, 2007Co-Authors: Jenő Szigeti, Leon Van WykAbstract:Abstract Let S be a subRing of the Ring R. We investigate the question of whether S ∩ U ( R ) = U ( S ) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full Matrix Ring and S is a structural subRing of R defined by a reflexive and transitive relation.
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subRings which are closed with respect to taking the inverse
arXiv: Rings and Algebras, 2007Co-Authors: Jenő Szigeti, Leon Van WykAbstract:Let S be a subRing of the Ring R. We investigate the question of whether S intersected by U(R) is equal to U(S) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full Matrix Ring and S is a structural subRing of R defined by a reflexive and transitive relation.
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Matrix Rings satisfying column sum conditions versus structural Matrix Rings
Linear Algebra and its Applications, 1996Co-Authors: Leon Van WykAbstract:Abstract The internal characterization of a structural Matrix Ring in terms of a set of Matrix units associated with a quasiorder relation is used to obtain isomorphisms between seemingly different classes of subRings of a complete Matrix Ring.
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On a characterization of complete Matrix Rings
Periodica Mathematica Hungarica, 1996Co-Authors: Leon Van WykAbstract:Peter R. Fuchs established in 1991 a new characterization of complete Matrix Rings by showing that a RingR with identity is isomorphic to a Matrix RingMn(S) for some RingS (and somen ≥ 2) if and only if there are elementsx andy inR such thatxn−1 ≠ 0,xn=0=y2,x+y is invertible, and Ann(xn−1)∩Ry={0} (theintersection condition), and he showed that the intersection condition is superfluous in casen=2. We show that the intersection condition cannot be omitted from Fuchs' characterization ifn≥3; in fact, we show that if the intersection condition is omitted, then not only may it happen that we do not obtain a completen ×n Matrix Ring for then under consideration, but it may even happen that we do not obtain a completem ×m Matrix Ring for anym≥2.
Daniel Eremita - One of the best experts on this subject based on the ideXlab platform.
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Commuting traces of upper triangular Matrix Rings
Aequationes Mathematicae, 2017Co-Authors: Daniel EremitaAbstract:Let \(T_n(R)\) be the upper triangular Matrix Ring over a unital Ring R. Suppose that \(B:T_n(R)\times T_n(R) \rightarrow T_n(R)\) is a biadditive map such that \(B(X,X)X = XB(X,X)\) for all \(X \in T_n(R)\). We consider the problem of describing the form of the map \(X\mapsto B(X,X)\).
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functional identities in upper triangular Matrix Rings
Linear Algebra and its Applications, 2016Co-Authors: Daniel EremitaAbstract:Abstract Let R be a subRing of a Ring Q, both having the same unity. We prove that if R is a d-free subset of Q, then the upper triangular Matrix Ring T n ( R ) is a d-free subset of T n ( Q ) for any n ∈ N .
