1. CMB 2017 (vol 60 pp. 791)
 Jiang, Chunlan

Reduction to Dimension Two of Local Spectrum for $AH$ Algebra with Ideal Property
A $C^{*}$algebra $A$ has the ideal property if any ideal
$I$ of $A$ is generated as a closed two sided ideal by the projections
inside the ideal. Suppose that the limit $C^{*}$algebra $A$
of inductive limit of direct sums of matrix algebras over spaces
with uniformly bounded dimension has ideal property. In this
paper we will prove that $A$ can be written as an inductive limit
of certain very special subhomogeneous algebras, namely, direct
sum of dimension drop interval algebras and matrix algebras over
2dimensional spaces with torsion $H^{2}$ groups.
Keywords:AH algebra, reduction, local spectrum, ideal property Category:46L35 

2. CMB 2017 (vol 60 pp. 372)
 Rao, M. Sambasiva

Coaxer Lattices
The notion of coaxers is introduced in a pseudocomplemented
distributive lattice. Boolean algebras are characterized in terms
of coaxer ideals and congruences. The concept of coaxer lattices
is introduced in pseudocomplemented distributive lattices and
characterized in terms of coaxer ideals and maximal ideals. Finally,
the coaxer lattices are also characterized in topological terms.
Keywords:pseudocomplemented distributive lattice, coaxer ideal, coaxer lattice, maximal ideal, congruence, kernel, antikernel Category:06D99 

3. CMB 2016 (vol 60 pp. 3)
 Akbari, Saeeid; Alilou, Abbas; Amjadi, Jafar; Sheikholeslami, Seyed Mahmoud

The Coannihilatingideal 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 

4. CMB 2016 (vol 60 pp. 63)
 Chang, Gyu Whan

Power Series Rings Over PrÃ¼fer $v$multiplication Domains, II
Let $D$ be an integral domain, $X^1(D)$ be the set of heightone
prime ideals of $D$,
$\{X_{\beta}\}$ and $\{X_{\alpha}\}$ be
two disjoint nonempty sets of indeterminates over $D$,
$D[\{X_{\beta}\}]$ be the polynomial ring over $D$, and
$D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1$ be the first type
power series ring over $D[\{X_{\beta}\}]$.
Assume that $D$ is a PrÃ¼fer $v$multiplication domain (P$v$MD)
in which each proper integral $t$ideal has only finitely many
minimal prime ideals
(e.g., $t$SFT P$v$MDs, valuation domains, rings of Krull type).
Among other things, we show that if
$X^1(D) = \emptyset$ or $D_P$ is a DVR for all $P \in X^1(D)$,
then
${D[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1}_{D  \{0\}}$ is a
Krull domain.
We also prove that if $D$ is a $t$SFT P$v$MD, then the complete
integral closure of $D$ is a Krull domain and
ht$(M[\{X_{\beta}\}][\![\{X_{\alpha}\}]\!]_1)$ = $1$ for every
heightone maximal $t$ideal $M$ of $D$.
Keywords:Krull domain, P$v$MD, multiplicatively closed set of ideals, power series ring Categories:13A15, 13F05, 13F25 

5. CMB 2016 (vol 59 pp. 641)
 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 

6. CMB 2016 (vol 59 pp. 244)
 Cao, Wensheng; Huang, Xiaolin

A Note on Quaternionic Hyperbolic Ideal Triangle Groups
In this paper, the quaternionic hyperbolic
ideal triangle groups are parameterized by a real oneparameter
family $\{\phi_s: s\in \mathbb{R}\}$. The indexing parameter $s$ is
the tangent of the quaternionic angular invariant of a triple
of points in $\partial \mathbf{H}_{\mathbb{h}}^2 $ forming this ideal
triangle. We show that if $s \gt \sqrt{125/3}$ then $\phi_s$ is
not a discrete embedding, and if $s \leq \sqrt{35}$
then $\phi_s$ is a discrete embedding.
Keywords:quaternionic inversion, ideal triangle group, quaternionic Cartan angular invariant Categories:20F67, 22E40, 30F40 

7. 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 

8. CMB 2015 (vol 58 pp. 350)
 MerinoCruz, Héctor; Wawrzynczyk, Antoni

On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions
We recently introduced a weighted Banach algebra $\mathfrak{A}_G^n$ of
functions which are holomorphic on the unit disc $\mathbb{D}$, continuous
up to the boundary and of the class $C^{(n)}$ at all points where
the function $G$ does not vanish. Here, $G$ refers to a function
of the disc algebra without zeros on $\mathbb{D}$. Then we proved that
all closed ideals in $\mathfrak{A}_G^n$ with at most countable hull are
standard. In the present paper, on the assumption that $G$ is
an outer function in $C^{(n)}(\overline{\mathbb{D}})$ having infinite roots
in $\mathfrak{A}_G^n$ and countable zero set $h(G)$, we show that all the
closed ideals $I$ with hull containing $h(G)$ are standard.
Keywords:Banach algebra, disc algebra, holomorphic spaces, standard ideal Categories:46J15, 46J20, 30H50 

9. 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 

10. 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 

11. CMB 2014 (vol 57 pp. 780)
 Erzakova, Nina A.

Measures of Noncompactness in Regular Spaces
Previous results by the author on the connection
between three of measures
of noncompactness obtained for $L_p$, are extended
to regular spaces of measurable
functions.
An example of advantage
in some cases one of them in comparison with another is given.
Geometric characteristics of regular spaces are determined.
New theorems for $(k,\beta)$boundedness of partially additive
operators are proved.
Keywords:measure of noncompactness, condensing map, partially additive operator, regular space, ideal space Categories:47H08, 46E30, 47H99, 47G10 

12. CMB 2012 (vol 57 pp. 90)
 Lazar, Aldo J.

Compact Subsets of the Glimm Space of a $C^*$algebra
If $A$ is a $\sigma$unital $C^*$algebra and $a$ is a strictly positive element of $A$ then for every compact subset $K$ of the complete
regularization $\mathrm{Glimm}(A)$ of $\mathrm{Prim}(A)$ there exists
$\alpha \gt 0$ such that $K\subset \{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq
\alpha\}$. This extends
a result of J. Dauns
to all $\sigma$unital $C^*$algebras. However, there are a $C^*$algebra $A$
and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq
\alpha\}$, $a\in A$ and $\alpha \gt 0$.
Keywords:primitive ideal space, complete regularization Category:46L05 

13. CMB 2011 (vol 56 pp. 442)
 Zelenyuk, Yevhen

Closed Left Ideal Decompositions of $U(G)$
Let $G$ be an infinite discrete group and let $\beta G$ be the
StoneÄech compactification of $G$. We take the points of $Äta
G$ to be the ultrafilters on $G$, identifying the principal
ultrafilters with the points of $G$. The set $U(G)$ of uniform
ultrafilters on $G$ is a closed twosided ideal of $\beta G$. For
every $p\in U(G)$, define $I_p\subseteq\beta G$ by $I_p=\bigcap_{A\in
p}\operatorname{cl} (GU(A))$, where $U(A)=\{p\in U(G):A\in p\}$. We show
that if $G$ is a regular cardinal, then $\{I_p:p\in U(G)\}$ is the
finest decomposition of $U(G)$ into closed left ideals of $\beta G$
such that the corresponding quotient space of $U(G)$ is Hausdorff.
Keywords:StoneÄech compactification, uniform ultrafilter, closed left ideal, decomposition Categories:22A15, 54H20, 22A30, 54D80 

14. CMB 2011 (vol 56 pp. 434)
 Wnuk, Witold

Some Remarks on the Algebraic Sum of Ideals and Riesz Subspaces
Following ideas used by Drewnowski and Wilansky we prove that if $I$
is an infinite dimensional and
infinite codimensional closed ideal in a complete metrizable locally
solid Riesz space and $I$ does
not contain any order copy of $\mathbb R^{\mathbb N}$ then there exists a
closed, separable, discrete Riesz subspace
$G$ such that the topology induced on $G$ is Lebesgue, $I \cap G =
\{0\}$, and $I + G$ is not closed.
Keywords:locally solid Riesz space, Riesz subspace, ideal, minimal topological vector space, Lebesgue property Categories:46A40, 46B42, 46B45 

15. CMB 2011 (vol 54 pp. 487)
 Kong, Xiangjun

Some Properties Associated with Adequate Transversals
In this paper, another relationship between the quasiideal adequate transversals
of an abundant semigroup is given. We introduce the concept of a weakly multiplicative
adequate transversal and the classic result that an adequate transversal is multiplicative
if and only if it is weakly multiplicative and a quasiideal is obtained.
Also, we give two equivalent conditions for an adequate transversal to be weakly
multiplicative. We then consider the case when $I$ and $\Lambda$ (defined below) are
bands. This is analogous to the inverse transversal if the regularity condition is adjoined.
Keywords:abundant semigroup, adequate transversal, Green's $*$relations, quasiideal Category:20M10 

16. CMB 2010 (vol 53 pp. 690)
 Puerta, M. E.; Loaiza, G.

On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces
The classical approach to studying operator ideals using tensor
norms mainly focuses on those tensor norms and operator ideals
defined by means of $\ell_p$ spaces. In a previous paper,
an interpolation space, defined via the real method
and using
$\ell_p$ spaces, was used to define a tensor
norm, and the associated minimal operator ideals were characterized.
In this paper, the next natural step is taken, that is, the
corresponding maximal operator
ideals are characterized. As an application, necessary and sufficient
conditions for the coincidence of
the maximal and minimal ideals are given.
Finally, the previous results are used in order to find some new
metric properties of the mentioned tensor norm.
Keywords:maximal operator ideals, ultraproducts of spaces, interpolation spaces Categories:46M05, 46M35, 46A32 

17. CMB 2010 (vol 53 pp. 577)
18. CMB 2005 (vol 48 pp. 121)
19. CMB 2001 (vol 44 pp. 504)
 Zhang, Yong

Weak Amenability of a Class of Banach Algebras
We show that, if a Banach algebra $\A$ is a left ideal in its second
dual algebra and has a left bounded approximate identity, then the
weak amenability of $\A$ implies the ($2m+1$)weak amenability of $\A$
for all $m\geq 1$.
Keywords:$n$weak amenability, left ideals, left bounded approximate identity Categories:46H20, 46H10, 46H25 

20. CMB 1999 (vol 42 pp. 118)
 Rao, T. S. S. R. K.

Points of Weak$^\ast$Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$
For a compact Hausdorff space with a dense set of isolated points, we
give a complete description of points of weak$^\ast$norm continuity
in the dual unit ball of the space of Banach space valued functions
that are continuous when the range has the weak topology. As an
application we give a complete description of points of weaknorm
continuity of the unit ball of the space of vector measures when
the underlying Banach space has the RadonNikodym property.
Keywords:Points of weak$^\ast$norm continuity, space of vector valued weakly continuous functions, $M$ideals Categories:46B20, 46E40 

21. CMB 1997 (vol 40 pp. 309)
 Hillman, J. A.; Sakuma, M.

On the homology of finite abelian coverings of links
Let $A$ be a finite abelian group and $M$ be a
branched cover of an homology $3$sphere, branched over a link $L$,
with covering group $A$. We show that $H_1(M;Z[1/A])$ is determined
as a $Z[1/A][A]$module by the Alexander ideals of $L$ and certain
ideal class invariants.
Keywords:Alexander ideal, branched covering, Dedekind domain,, knot, link. Category:57M25 

22. CMB 1997 (vol 40 pp. 47)
 Hartl, Manfred

A universal coefficient decomposition for subgroups induced by submodules of group algebras
Dimension subgroups and Lie dimension subgroups are known to satisfy a
`universal coefficient decomposition', {\it i.e.} their value with respect to
an arbitrary coefficient ring can be described in terms of their values with
respect to the `universal' coefficient rings given by the cyclic groups of
infinite and prime power order. Here this fact is generalized to much more
general types of induced subgroups, notably covering Fox subgroups and
relative dimension subgroups with respect to group algebra filtrations
induced by arbitrary $N$series, as well as certain common generalisations
of these which occur in the study of the former. This result relies on an
extension of the principal universal coefficient decomposition theorem on
polynomial ideals (due to Passi, Parmenter and Seghal), to all additive
subgroups of group rings. This is possible by using homological instead
of ring theoretical methods.
Keywords:induced subgroups, group algebras, Fox subgroups, relative dimension, subgroups, polynomial ideals Categories:20C07, 16A27 
