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Search: All articles in the CMB digital archive with keyword nilpotent

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1. CMB Online first

Lee, Tsiu-Kwen
Ad-nilpotent elements of semiprime rings with involution
Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\gt 1$ is a positive integer. We characterize $a\in K$, the Lie algebra of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This generalizes both Martindale and Miers' theorem and the theorem of Brox et al. To prove it we first prove that if $a, b\in R$ satisfy $(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on $R$, where either $n$ is even or $b=0$, then $\big(a-\lambda\big)^{[\frac{n+1}{2}]}=0$ for some $\lambda\in C$.

Keywords:Semiprime ring, Lie algebra, Jordan algebra, faithful $f$-free, involution, skew (symmetric) element, ad-nilpotent element, Jordan element
Categories:16N60, 16W10, 17B60

2. CMB Online first

Khojasteh, Sohiela; Nikmehr, Mohammad Javad
The Weakly Nilpotent Graph of a Commutative Ring
Let $R$ be a commutative ring with non-zero identity. In this paper, we introduced the weakly nilpotent graph of a commutative ring. The weakly nilpotent graph of $R$ is denoted by $\Gamma_w(R)$ is a graph with the vertex set $R^{*}$ and two vertices $x$ and $y$ are adjacent if and only if $xy\in N(R)^{*}$, where $R^{*}=R\setminus\{0\}$ and $N(R)^{*}$ is the set of all non-zero nilpotent elements of $R$. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if $\Gamma_w(R)$ is a forest, then $\Gamma_w(R)$ is a union of a star and some isolated vertices. We study the clique number, the chromatic number and the independence number of $\Gamma_w(R)$. Among other results, we show that for an Artinian ring $R$, $\Gamma_w(R)$ is not a disjoint union of cycles or a unicyclic graph. For Artinan ring, we determine $\operatorname{diam}(\overline{\Gamma_w(R)})$. Finally, we characterize all commutative rings $R$ for which $\overline{\Gamma_w(R)}$ is a cycle, where $\overline{\Gamma_w(R)}$ is the complement of the weakly nilpotent graph of $R$.

Keywords:weakly nilpotent graph, zero-divisor graph, diameter, girth
Categories:05C15, 16N40, 16P20

3. CMB 2016 (vol 60 pp. 12)

Akbari, Saieed; Miraftab, Babak; Nikandish, Reza
Co-maximal Graphs of Subgroups of Groups
Let $H$ be a group. The co-maximal graph of subgroups of $H$, denoted by $\Gamma(H)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices $L$ and $K$ are adjacent in $\Gamma(H)$ if and only if $H=LK$. In this paper, we study the connectivity, diameter, clique number and vertex chromatic number of $\Gamma(H)$. For instance, we show that if $\Gamma(H)$ has no isolated vertex, then $\Gamma(H)$ is connected with diameter at most $3$. Also, we characterize all finite groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma(H)$ is connected and moreover the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.

Keywords:co-maximal graphs of subgroups of groups, diameter, nilpotent group, solvable group
Categories:05C25, 05E15, 20D10, 20D15

4. CMB 2016 (vol 60 pp. 77)

Christ, Michael; Rieffel, Marc A.
Nilpotent Group C*-algebras as Compact Quantum Metric Spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a ``compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth
Categories:46L87, 20F65, 22D15, 53C23, 58B34

5. CMB 2016 (vol 59 pp. 461)

Ara, Pere; O'Meara, Kevin C.
The Nilpotent Regular Element Problem
We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular elements in exchange rings (a large class of rings), and for general rings when all powers of the nilpotent element $x$ are regular.

Keywords:nilpotent element, von Neumann regular element, unit-regular, Bergman's normal form
Categories:16E50, 16U99, 16S10, 16S15

6. CMB 2014 (vol 57 pp. 884)

Xu, Yong; Zhang, Xinjian
$m$-embedded Subgroups and $p$-nilpotency of Finite Groups
Let $A$ be a subgroup of a finite group $G$ and $\Sigma : G_0\leq G_1\leq\cdots \leq G_n$ some subgroup series of $G$. Suppose that for each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and $G_{i-1}\leq K \lt H\leq G_i$, for some $i$, either $A\cap H = A\cap K$ or $AH = AK$. Then $A$ is said to be $\Sigma$-embedded in $G$; $A$ is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup $T$ and a $\{1\leq G\}$-embedded subgroup $C$ in $G$ such that $G = AT$ and $T\cap A\leq C\leq A$. In this article, some sufficient conditions for a finite group $G$ to be $p$-nilpotent are given whenever all subgroups with order $p^{k}$ of a Sylow $p$-subgroup of $G$ are $m$-embedded for a given positive integer $k$.

Keywords:finite group, $p$-nilpotent group, $m$-embedded subgroup
Categories:20D10, 20D15

7. CMB 2013 (vol 57 pp. 125)

Mlaiki, Nabil M.
Camina Triples
In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs. Camina pairs were first introduced in 1978 by A .R. Camina. Camina's work was inspired by the study of Frobenius groups. We show that if $(G,N,M)$ is a Camina triple, then either $G/N$ is a $p$-group, or $M$ is abelian, or $M$ has a non-trivial nilpotent or Frobenius quotient.

Keywords:Camina triples, Camina pairs, nilpotent groups, vanishing off subgroup, irreducible characters, solvable groups

8. CMB 2012 (vol 56 pp. 606)

Mazorchuk, Volodymyr; Zhao, Kaiming
Characterization of Simple Highest Weight Modules
We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.

Keywords:Lie algebra, highest weight module, triangular decomposition, locally nilpotent action
Categories:17B20, 17B65, 17B66, 17B68

9. CMB 2011 (vol 55 pp. 579)

Ndogmo, J. C.
Casimir Operators and Nilpotent Radicals
It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established.

Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariants
Categories:16W25, 17B45, 16S30

10. CMB 2009 (vol 52 pp. 535)

Daigle, Daniel; Kaliman, Shulim
A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$
We strengthen certain results concerning actions of $(\Comp,+)$ on $\Comp^{3}$ and embeddings of $\Comp^{2}$ in $\Comp^{3}$, and show that these results are in fact valid over any field of characteristic zero.

Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine space
Categories:14R10, 14R20, 14R25, 13N15

11. CMB 2004 (vol 47 pp. 343)

Drensky, Vesselin; Hammoudi, Lakhdar
Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras
We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite words
Categories:16N40, 16S15, 20M05, 20M25, 68R15

12. CMB 2001 (vol 44 pp. 266)

Cencelj, M.; Dranishnikov, A. N.
Extension of Maps to Nilpotent Spaces
We show that every compactum has cohomological dimension $1$ with respect to a finitely generated nilpotent group $G$ whenever it has cohomological dimension $1$ with respect to the abelianization of $G$. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum $X$ for extendability of every map from a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely generated homotopy groups over all of $X$.

Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent space
Categories:55M10, 55S36, 54C20, 54F45

13. CMB 1999 (vol 42 pp. 335)

Kim, Goansu; Tang, C. Y.
Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups
We derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of HNN-extensions of nilpotent groups with cyclic associated subgroups.

Keywords:HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite
Categories:20E26, 20E06, 20F10

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