1. CMB Online first
 Khojasteh, Sohiela; Nikmehr, Mohammad Javad

The Weakly Nilpotent Graph of a Commutative Ring
Let $R$ be a commutative ring with nonzero 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 nonzero 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, zerodivisor graph, diameter, girth Categories:05C15, 16N40, 16P20 

2. CMB 2016 (vol 60 pp. 12)
 Akbari, Saieed; Miraftab, Babak; Nikandish, Reza

Comaximal Graphs of Subgroups of Groups
Let $H$ be a group. The comaximal graph of subgroups
of $H$, denoted by $\Gamma(H)$, is a
graph whose vertices are nontrivial 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 comaximal 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
comaximal graph of a general linear group over an algebraically
closed field is zero or infinite.
Keywords:comaximal graphs of subgroups of groups, diameter, nilpotent group, solvable group Categories:05C25, 05E15, 20D10, 20D15 

3. 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 wordlength functions
on finitely generated nilpotentbyfinite groups.
Keywords:group C*algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth Categories:46L87, 20F65, 22D15, 53C23, 58B34 

4. 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 unitregular.
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, unitregular, Bergman's normal form Categories:16E50, 16U99, 16S10, 16S15 

5. 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
$G_{i1}\leq K \lt H\leq G_i$, 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 

6. 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 nontrivial nilpotent or
Frobenius quotient.
Keywords:Camina triples, Camina pairs, nilpotent groups, vanishing off subgroup, irreducible characters, solvable groups Category:20D15 

7. 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 KacMoody Lie algebras, the
Virasoro algebra and the HeisenbergVirasoro 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 

8. 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 

9. 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 

10. 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 nonnil 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 

11. 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 finitedimensional 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 

12. CMB 1999 (vol 42 pp. 335)
 Kim, Goansu; Tang, C. Y.

Cyclic Subgroup Separability of HNNExtensions with Cyclic Associated Subgroups
We derive a necessary and sufficient condition for HNNextensions
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 HNNextensions of abelian groups with
cyclic associated subgroups. We also consider these residual
properties of HNNextensions of nilpotent groups with cyclic
associated subgroups.
Keywords:HNNextension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite Categories:20E26, 20E06, 20F10 
