1. CMB Online first
 Shaveisi, Farzad

Some Results on the AnnihilatingIdeal Graphs
The annihilatingideal graph
of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, is a
graph whose vertex set consists of all nonzero annihilating
ideals and two distinct
vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Here,
we show that if $R$ is a reduced ring and the independence
number of $\mathbb{AG}(R)$ is finite, then the edge chromatic
number of $\mathbb{AG}(R)$ equals its maximum degree
and this number equals $2^{{\rm Min}(R)1}1$; also, it is
proved that the independence number of $\mathbb{AG}(R)$ equals
$2^{{\rm Min}(R)1}$, where ${\rm Min}(R)$ denotes the set
of minimal prime ideals of $R$.
Then we give some criteria for a graph to be isomorphic with
an annihilatingideal graph of a ring.
For example, it is shown that every bipartite annihilatingideal
graph is a complete bipartite graph with at most two horns. Among
other results, it is shown that a finite graph $\mathbb{AG}(R)$
is not Eulerian, and it is Hamiltonian if and only if $R$ contains
no Gorenstain ring as its direct summand.
Keywords:annihilatingideal graph, independence number, edge chromatic number, bipartite, cycle Categories:05C15, 05C69, 13E05, 13E10 

2. CMB Online first
 Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei

Canonical systems of basic invariants for unitary reflection groups
It has been known that there exists a canonical system for every
finite real reflection group. The first and the third authors
obtained
an explicit formula for a canonical system in the previous paper.
In this article, we first define canonical systems for the finite
unitary reflection groups, and then prove their existence.
Our proof does not depend on the classification of unitary reflection
groups.
Furthermore, we give an explicit formula for a canonical system
for every unitary reflection group.
Keywords:basic invariant, invariant theory, finite unitary reflection group Categories:13A50, 20F55 

3. CMB Online first
 Dolžan, David

The metric dimension of the total graph of a finite commutative ring
We study the total graph of a finite commutative ring. We calculate
its metric dimension in the case when the Jacobson radical of
the ring is nontrivial and we examine the metric dimension of
the total graph of a product of at most two fields, obtaining
either exact values in some cases or bounds in other, depending
on the number of elements in the respective fields.
Keywords:total graph, finite ring, metric dimension Categories:13M99, 05E40 

4. CMB Online first
 Akbari, Saeeid; Alilou, Abbas; Amjadi, Jafar; Sheikholeslami, Seyed Mahmoud

The coannihilating ideal graphs of commutative rings
Let $R$ be a commutative ring with identity. The
coannihilatingideal graph of $R$, denoted by $\mathcal{A}_R$,
is
a graph whose vertex set is the set of all nonzero proper ideals
of $R$ and two distinct vertices $I$ and $J$ are adjacent
whenever ${\operatorname {Ann}}(I)\cap {\operatorname {Ann}}(J)=\{0\}$. In this paper we
initiate the study of the coannihilating ideal graph of a
commutative ring and we investigate its properties.
Keywords:commutative ring, coannihilating ideal graph Categories:13A15, 16N40 

5. CMB 2016 (vol 59 pp. 271)
 DehghaniZadeh, Fatemeh

Artinianness of Composed Graded Local Cohomology Modules
Let $R=\bigoplus_{n\geq0}R_{n}$ be a graded Noetherian ring with
local base ring $(R_{0}, \mathfrak{m}_{0})$ and let
$R_{+}=\bigoplus_{n\gt 0}R_{n}$, $M$ and $N$ be finitely generated
graded $R$modules and $\mathfrak{a}=\mathfrak{a}_{0}+R_{+}$ an ideal of $R$. We
show that $H^{j}_{\mathfrak{b}_{0}}(H^{i}_{\mathfrak{a}}(M,N))$ and $H^{i}_{\mathfrak{a}}(M,
N)/\mathfrak{b}_{0}H^{i}_{\mathfrak{a}}(M,N)$ are Artinian for some $i^{,}s$ and
$j^{,}s$ with a specified property, where $\mathfrak{b}_{o}$ is an ideal
of
$R_{0}$ such that $\mathfrak{a}_{0}+\mathfrak{b}_{0}$ is an $\mathfrak{m}_{0}$primary ideal.
Keywords:generalized local cohomology, Artinian, graded module Categories:13D45, 13E10, 16W50 

6. CMB 2016 (vol 59 pp. 403)
 Zargar, Majid Rahro; Zakeri, Hossein

On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules
Let $\mathfrak{a}$ be an ideal of a Noetherian local
ring $R$ and let $C$ be a semidualizing $R$module. For an $R$module
$X$, we denote any of the quantities $\mathfrak{d}_R X$,
$\operatorname{\mathsf{Gfd}}_R X$ and
$\operatorname{\mathsf{G_Cfd}}_RX$ by $\operatorname{\mathsf{T}}(X)$. Let $M$ be an $R$module such that
$\operatorname{H}_{\mathfrak{a}}^i(M)=0$
for all $i\neq n$. It is proved that if $\operatorname{\mathsf{T}}(X)\lt \infty$, then
$\operatorname{\mathsf{T}}(\operatorname{H}_{\mathfrak{a}}^n(M))\leq\operatorname{\mathsf{T}}(M)+n$ and the equality holds whenever
$M$ is finitely generated. With the aid of these results, among
other things, we characterize CohenMacaulay modules, dualizing
modules and Gorenstein rings.
Keywords:flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative CohenMacaulay module, semidualizing module Categories:13D05, 13D45, 18G20 

7. CMB 2015 (vol 59 pp. 197)
 Rajaee, Saeed

Quasicopure Submodules
All rings are commutative with identity and all modules are unital.
In this paper we introduce the concept of quasicopure submodule
of
a multiplication $R$module $M$ and will give some results of
them.
We give some properties of tensor product of finitely generated
faithful multiplication modules.
Keywords:multiplication module, arithmetical ring, copure submodule, radical of submodules Categories:13A15, 13C05, 13C13, , 13C99 

8. CMB 2015 (vol 58 pp. 704)
 Benamar, H.; Chandoul, A.; Mkaouar, M.

On the Continued Fraction Expansion of Fixed Period in Finite Fields
The Chowla conjecture
states that,
if $t$ is any given
positive integer, there are infinitely many prime positive
integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where
$\operatorname{Per} (\sqrt{N})$
is the period length of the continued fraction expansion for
$\sqrt{N}$.
C. Friesen proved
that, for any $k\in \mathbb{N}$, there are infinitely many
squarefree integers $N$, where the continued fraction expansion
of $\sqrt{N}$ has a fixed period. In this paper, we describe all
polynomials $Q\in \mathbb{F}_q[X] $ for which the continued fraction
expansion of $\sqrt {Q}$ has a fixed period, also we give a
lower
bound of the number of monic, nonsquares polynomials $Q$ such
that $\deg Q= 2d$ and $ Per \sqrt {Q}=t$.
Keywords:continued fractions, polynomials, formal power series Categories:11A55, 13J05 

9. CMB 2015 (vol 58 pp. 664)
 Vahidi, Alireza

Betti Numbers and Flat Dimensions of Local Cohomology Modules
Assume that $R$ is a commutative Noetherian ring with nonzero
identity, $\mathfrak{a}$ is an ideal of $R$ and $X$ is an $R$module.
In this paper, we first study the finiteness of Betti numbers
of local cohomology modules $\operatorname{H}_\mathfrak{a}^i(X)$. Then we give some
inequalities between the Betti numbers of $X$ and those of its
local cohomology modules. Finally, we present many upper bounds
for the flat dimension of $X$ in terms of the flat dimensions
of its local cohomology modules and an upper bound for the flat
dimension of $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of
the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$.
Keywords:Betti numbers, flat dimensions, local cohomology modules Categories:13D45, 13D05 

10. CMB 2015 (vol 58 pp. 449)
 Boynton, Jason Greene; Coykendall, Jim

On the Graph of Divisibility of an Integral Domain
It is well known that the factorization properties of a domain are reflected
in the structure of its group of divisibility. The main theme of this paper
is to introduce a topological/graphtheoretic point of view to the current
understanding of factorization in integral domains. We also show that
connectedness properties in the graph and topological space give rise to a
generalization of atomicity.
Keywords:atomic, factorization, divisibility Categories:13F15, 13A05 

11. CMB 2015 (vol 58 pp. 320)
 Llamas, Aurora; MartínezBernal, José

Cover Product and Betti Polynomial of Graphs
For disjoint graphs $G$ and $H$, with fixed
vertex covers
$C(G)$ and $C(H)$, their cover product is the graph $G
\circledast
H$ with vertex set
$V(G)\cup V(H)$ and edge set $E(G)\cup E(H)\cup\{\{i,j\}:i\in
C(G), j\in
C(H)\}$. We describe the graded Betti numbers of $G\circledast
H$ in terms of those of
$G$ and $H$. As applications we obtain: (i) For any positive
integer $k$ there
exists a connected bipartite graph $G$ such that $\operatorname{reg}
R/I(G)=\mu_S(G)+k$, where,
$I(G)$ denotes the edge ideal of $G$, $\operatorname{reg} R/I(G)$
is the CastelnuovoMumford
regularity of $R/I(G)$ and $\mu_S(G)$ is the induced or strong
matching number of
$G$; (ii) The graded Betti numbers of the complement of a tree
only depends upon
its number of vertices; (iii) The $h$vector of $R/I(G\circledast
H)$ is described in
terms of the $h$vectors of $R/I(G)$ and $R/I(H)$. Furthermore,
in a different
direction, we give a recursive formula for the graded Betti numbers
of chordal
bipartite graphs.
Keywords:CastelnuovoMumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial ideal Categories:13D02, 05E45 

12. CMB 2015 (vol 58 pp. 393)
 Tang, Zhongming

On Stanley Depths of Certain Monomial Factor Algebras
Let $S=K[x_1,\ldots,x_n]$
be the polynomial
ring in $n$variables over a field $K$ and $I$ a monomial ideal
of $S$. According to one standard primary decomposition of $I$,
we get a Stanley decomposition of the monomial factor algebra
$S/I$.
Using this Stanley decomposition, one can estimate the Stanley
depth of $S/I$. It is proved that
${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {size}}_S(I)$. When $I$ is squarefree
and ${\operatorname {bigsize}}_S(I)\leq 2$, the Stanley conjecture holds
for
$S/I$, i.e., ${\operatorname {sdepth}}_S(S/I)\geq{\operatorname {depth}}_S(S/I)$.
Keywords:monomial ideal, size, Stanley depth Categories:13F20, 13C15 

13. CMB 2014 (vol 58 pp. 134)
14. CMB Online first
15. CMB 2014 (vol 57 pp. 573)
 Kiani, Sima; Maimani, Hamid Reza; Nikandish, Reza

Some Results on the Domination Number of a Zerodivisor Graph
In this paper, we investigate the domination, total domination and
semitotal domination numbers of a zerodivisor graph of a
commutative Noetherian ring. Also, some relations between the
domination numbers of $\Gamma(R/I)$ and $\Gamma_I(R)$, and the
domination numbers of $\Gamma(R)$ and $\Gamma(R[x,\alpha,\delta])$,
where $R[x,\alpha,\delta]$ is the Ore extension of $R$, are studied.
Keywords:zerodivisor graph, domination number Categories:05C75, 13H10 

16. CMB 2014 (vol 57 pp. 477)
 Eghbali, Majid

On Set Theoretically and Cohomologically Complete Intersection Ideals
Let $(R,\mathfrak m)$ be a local ring and $\mathfrak a$ be an ideal of $R$. The inequalities
\[
\operatorname{ht}(\mathfrak a) \leq \operatorname{cd}(\mathfrak a,R) \leq
\operatorname{ara}(\mathfrak a) \leq
l(\mathfrak a) \leq \mu(\mathfrak a)
\]
are known. It is an interesting and longstanding problem to find
out the cases giving equality. Thanks to the formal grade we give
conditions in which the above inequalities become
equalities.
Keywords:settheoretically and cohomologically complete intersection ideals, analytic spread, monomials, formal grade, depth of powers of ideals Categories:13D45, 13C14 

17. CMB 2014 (vol 57 pp. 735)
 Cagliero, Leandro; Szechtman, Fernando

On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras
We describe of all finite
dimensional uniserial representations of a commutative associative
(resp. abelian Lie) algebra over a perfect (resp. sufficiently
large perfect) field. In the Lie case the size of the field
depends on the answer to following question, considered and solved
in this paper. Let $K/F$ be a finite separable field extension
and
let $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some
nonzero elements $\alpha,\beta\in F$?
Keywords:uniserial module, Lie algebra, associative algebra, primitive element Categories:17B10, 13C05, 12F10, 12E20 

18. CMB 2013 (vol 57 pp. 413)
 Samei, Karim

On the Comaximal Graph of a Commutative Ring
Let $R$ be a commutative ring with $1$. In [P. K. Sharma, S. M.
Bhatwadekar, A note on graphical representation of rings, J.
Algebra 176(1995) 124127], Sharma and Bhatwadekar defined a
graph on $R$, $\Gamma(R)$, with vertices as elements of $R$, where
two distinct vertices $a$ and $b$ are adjacent if and only if $Ra
+ Rb = R$. In this paper, we consider a subgraph $\Gamma_2(R)$ of
$\Gamma(R)$ which consists of nonunit elements. We investigate
the behavior of $\Gamma_2(R)$ and $\Gamma_2(R) \setminus \operatorname{J}(R)$,
where $\operatorname{J}(R)$ is the Jacobson radical of $R$. We associate the
ring properties of $R$, the graph properties of $\Gamma_2(R)$ and
the topological properties of $\operatorname{Max}(R)$. Diameter, girth, cycles
and dominating sets are investigated and the algebraic and the
topological characterizations are given for graphical properties
of these graphs.
Keywords:comaximal, Diameter, girth, cycles, dominating set Category:13A99 

19. CMB 2013 (vol 57 pp. 188)
20. CMB 2013 (vol 57 pp. 159)
 Oral, Kürşat Hakan; Özkirişci, Neslihan Ayşen; Tekir, Ünsal

Strongly $0$dimensional Modules
In a multiplication module, prime submodules have the property, if a prime
submodule contains a finite intersection of submodules then one of the
submodules is contained in the prime submodule. In this paper, we generalize
this property to infinite intersection of submodules and call such prime
submodules strongly prime submodule. A multiplication module in which every
prime submodule is strongly prime will be called strongly 0dimensional
module. It is also an extension of strongly 0dimensional rings. After
this we investigate properties of strongly 0dimensional modules and give
relations of von Neumann regular modules, Qmodules and strongly
0dimensional modules.
Keywords:strongly 0dimensional rings, Qmodule, Von Neumann regular module Categories:13C99, 16D10 

21. CMB 2012 (vol 57 pp. 289)
 Ghasemi, Mehdi; Marshall, Murray; Wagner, Sven

Closure of the Cone of Sums of $2d$powers in Certain Weighted $\ell_1$seminorm Topologies
In a paper from 1976, Berg, Christensen and Ressel prove that the
closure of the cone of sums of squares $\sum
\mathbb{R}[\underline{X}]^2$ in the polynomial ring
$\mathbb{R}[\underline{X}] := \mathbb{R}[X_1,\dots,X_n]$ in the
topology induced by the $\ell_1$norm is equal to
$\operatorname{Pos}([1,1]^n)$, the cone consisting of all polynomials
which are nonnegative on the hypercube $[1,1]^n$. The result is
deduced as a corollary of a general result, established in the same
paper, which is valid for any commutative semigroup.
In later work, Berg and Maserick and Berg, Christensen and Ressel
establish an even more general result, for a commutative semigroup
with involution, for the closure of the cone of sums of squares of
symmetric elements in the weighted $\ell_1$seminorm topology
associated to an absolute value.
In the present paper we give a new proof of these results which is
based on Jacobi's representation theorem from 2001. At the same time,
we use Jacobi's representation theorem to extend these results from
sums of squares to sums of $2d$powers, proving, in particular, that
for any integer $d\ge 1$, the closure of the cone of sums of
$2d$powers $\sum \mathbb{R}[\underline{X}]^{2d}$ in
$\mathbb{R}[\underline{X}]$ in the topology induced by the
$\ell_1$norm is equal to $\operatorname{Pos}([1,1]^n)$.
Keywords:positive definite, moments, sums of squares, involutive semigroups Categories:43A35, 44A60, 13J25 

22. CMB 2012 (vol 56 pp. 683)
 Nikseresht, A.; Azizi, A.

Envelope Dimension of Modules and the Simplified Radical Formula
We introduce and investigate the notion of envelope dimension of
commutative rings and modules over them. In particular, we show that
the envelope dimension of a ring, $R$, is equal to that of the
$R$module $R^{(\mathbb{N})}$. Also we prove that the Krull dimension of a
ring is no more than its envelope dimension and characterize
Noetherian rings for which these two dimensions are equal. Moreover we
generalize and study the concept of simplified radical formula for
modules, which
we defined in an earlier paper.
Keywords:envelope dimension, simplified radical formula, prime submodule Categories:13A99, 13C99, 13C13, 13E05 

23. CMB 2012 (vol 56 pp. 551)
 Handelman, David

Real Dimension Groups
Dimension groups (not countable) that are also real ordered vector
spaces can be obtained as direct limits (over directed sets) of
simplicial real vector spaces (finite dimensional vector spaces with
the coordinatewise ordering), but the directed set is not as
interesting as one would like, i.e., it is not true that a
countabledimensional real vector space that has interpolation can be
represented as such a direct limit over the a countable directed
set. It turns out this is the case when the group is additionally
simple, and it is shown that the latter have an ordered tensor product
decomposition. In the Appendix, we provide a huge class of polynomial
rings that, with a pointwise ordering, are shown to satisfy
interpolation, extending a result outlined by Fuchs.
Keywords:dimension group, simplicial vector space, direct limit, Riesz interpolation Categories:46A40, 06F20, 13J25, 19K14 

24. CMB 2012 (vol 56 pp. 534)
 Filali, M.; Monfared, M. Sangani

A Cohomological Property of $\pi$invariant Elements
Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$
be a continuous representation of $A$ on a separable Hilbert space $H$
with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of
$\pi$ with respect to an orthonormal basis and suppose that for each
$1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m}
\\pi_{ij}\_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these
conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$
left $\pi$invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all
$a\in A$. In this paper we prove a link between the existence
of left $\pi$invariant elements and the vanishing of certain
Hochschild cohomology groups of $A$. Our results extend an earlier
result by Lau on $F$algebras and recent results of KaniuthLauPym
and the second named author in the special case that $\pi \colon A
\longrightarrow \mathbf C$ is a nonzero character on $A$.
Keywords:Banach algebras, $\pi$invariance, derivations, representations Categories:46H15, 46H25, 13N15 

25. CMB 2012 (vol 56 pp. 491)
 Bahmanpour, Kamal

A Note on Homological Dimensions of Artinian Local Cohomology Modules
Let $(R,{\frak m})$ be a nonzero commutative Noetherian local ring
(with identity), $M$ be a nonzero finitely generated $R$module. In
this paper for any ${\frak p}\in {\rm Spec}(R)$ we show that
$
\operatorname{{\rm injdim_{_{R_{\frak p}}}}}
H^{i\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ and
${\rm fd}_{R_{\p}} H^{i\dim(R/{\frak p})}_{{\frak p}R_{\frak
p}}(M_{\frak p})$ are bounded from above by $
\operatorname{{\rm injdim_{_{R}}}}
H^i_{\frak
m}(M)$ and
$ {\rm fd}_R H^i_{\frak m}(M)$ respectively, for all integers $i\geq \dim(R/{\frak p})$.
Keywords:cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomology Category:13D45 
