26. 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 

27. CMB 2011 (vol 56 pp. 407)
 Rad, Nader Jafari; Jafari, Sayyed Heidar; Mojdeh, Doost Ali

On Domination in ZeroDivisor Graphs
We first determine the domination number for the zerodivisor
graph of the product of two commutative rings with $1$. We then
calculate the domination number for the zerodivisor graph of any
commutative artinian ring. Finally, we extend some of the results
to noncommutative rings in which an element is a left
zerodivisor if and only if it is a right zerodivisor.
Keywords:zerodivisor graph, domination Categories:13AXX, 05C69 

28. CMB 2011 (vol 56 pp. 31)
 Ayuso, Fortuny P.

Derivations and Valuation Rings
A complete characterization of valuation rings closed for a
holomorphic derivation is given, following an idea of Seidenberg,
in dimension $2$.
Keywords:singular holomorphic foliation, derivation, valuation, valuation ring Categories:32S65, 13F30, 13A18 

29. CMB 2011 (vol 55 pp. 752)
30. CMB 2011 (vol 55 pp. 378)
 Oman, Greg; Salminen, Adam

On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality
Let $R$ be a commutative ring with identity, and let $M$ be a
unitary module over $R$. We call $M$ Hsmaller (HS for short) if and only if
$M$ is infinite and $M/N<M$ for every nonzero submodule $N$ of
$M$. After a brief introduction, we show that there exist nontrivial
examples of HS modules of arbitrarily large cardinality over
Noetherian and nonNoetherian domains. We then prove the following
result: suppose $M$ is faithful over $R$, $R$ is a domain (we will
show that we can restrict to this case without loss of generality),
and $K$ is the quotient field of $R$. If $M$ is HS over $R$, then
$R$ is HS as a module over itself, $R\subseteq M\subseteq K$, and
there exists a generating set $S$ for $M$ over $R$ with $S<R$.
We use this result to generalize a problem posed by Kaplansky and
conclude the paper by answering an open question on JÃ³nsson
modules.
Keywords:Noetherian ring, residually finite ring, cardinal number, continuum hypothesis, valuation ring, JÃ³nsson module Categories:13A99, 13C05, 13E05, 03E50 

31. CMB 2011 (vol 55 pp. 127)
 LaGrange, John D.

Characterizations of Three Classes of ZeroDivisor Graphs
The zerodivisor graph $\Gamma(R)$ of a commutative ring $R$ is the graph whose vertices consist of
the nonzero zerodivisors of $R$ such that distinct vertices $x$ and
$y$ are adjacent if and only if $xy=0$. In this paper,
a characterization is provided for zerodivisor graphs of Boolean
rings. Also, commutative rings $R$ such that
$\Gamma(R)$ is isomorphic to the zerodivisor graph of a direct product of integral domains are classified, as well as
those whose zerodivisor graphs are central vertex complete.
Categories:13A99, 13M99 

32. CMB 2011 (vol 55 pp. 315)
 Hellus, M.

A Note on the Vanishing of Certain Local Cohomology Modules
For a finite module $M$ over a local, equicharacteristic ring $(R,m)$,
we show that the wellknown formula $\textrm{cd}(m,M)=\dim M$ becomes trivial
if ones uses Matlis duals of local cohomology modules together with spectral sequences.
We also prove a new ringtheoretic vanishing criterion for local cohomology modules.
Category:13D45 

33. CMB 2011 (vol 55 pp. 81)
34. CMB 2011 (vol 55 pp. 153)
 Mafi, Amir; Saremi, Hero

Artinianness of Certain Graded Local Cohomology Modules
We show that if
$R=\bigoplus_{n\in\mathbb{N}_0}R_n$ is a Noetherian homogeneous ring
with local base ring $(R_0,\mathfrak{m}_0)$, irrelevant ideal $R_+$, and
$M$ a finitely generated graded $R$module, then
$H_{\mathfrak{m}_0R}^j(H_{R_+}^t(M))$ is Artinian for $j=0,1$ where
$t=\inf\{i\in{\mathbb{N}_0}: H_{R_+}^i(M)$ is not finitely
generated $\}$. Also, we prove that if $\operatorname{cd}(R_+,M)=2$, then for
each $i\in\mathbb{N}_0$, $H_{\mathfrak{m}_0R}^i(H_{R_+}^2(M))$ is
Artinian if and only if $H_{\mathfrak{m}_0R}^{i+2}(H_{R_+}^1(M))$ is
Artinian, where $\operatorname{cd}(R_+,M)$ is the cohomological dimension of $M$
with respect to $R_+$. This improves some results of R. Sazeedeh.
Keywords:graded local cohomology, Artinian modules Categories:13D45, 13E10 

35. CMB 2011 (vol 54 pp. 619)
 Dibaei, Mohammad T.; Vahidi, Alireza

Artinian and NonArtinian Local Cohomology Modules
Let $M$ be a finite module over a commutative noetherian ring $R$.
For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between
cohomological dimensions of $M$ with respect to $\mathfrak{a},
\mathfrak{b}$,
$\mathfrak{a}\cap\mathfrak{b}$ and $\mathfrak{a}+ \mathfrak{b}$ are studied. When $R$ is local, it is
shown that $M$ is generalized CohenMacaulay if there exists an
ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with
respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer
such that $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the
nonempty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M)
$ is not artinian for
some $ i, i\geq r \}$ is a prime ideal, and all Bass numbers
of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$.
Keywords:local cohomology modules, cohomological dimensions, Bass numbers Categories:13D45, 13E10 

36. CMB 2011 (vol 54 pp. 716)
 Okassa, Eugène

Symplectic LieRinehartJacobi Algebras and Contact Manifolds
We give a characterization of contact manifolds in terms of symplectic
LieRinehartJacobi algebras. We also give a sufficient condition for a Jacobi
manifold to be a contact manifold.
Keywords:LieRinehart algebras, differential operators, Jacobi manifolds, symplectic manifolds, contact manifolds Categories:13N05, 53D05, 53D10 

37. CMB 2011 (vol 54 pp. 472)
 Iacono, Donatella

A Semiregularity Map Annihilating Obstructions to Deforming Holomorphic Maps
We study infinitesimal deformations of holomorphic maps of
compact, complex, KÃ¤hler manifolds. In particular, we describe a
generalization of Bloch's semiregularity map that annihilates
obstructions to deform holomorphic maps with fixed codomain.
Keywords:semiregularity map, obstruction theory, functors of Artin rings, differential graded Lie algebras Categories:13D10, 14D15, 14B10 

38. CMB 2010 (vol 53 pp. 639)
 Coykendall, Jim; Dutta, Tridib

A Generalization of Integrality
In this paper, we explore a generalization of the notion of
integrality. In particular, we study a nearintegrality condition that is
intermediate between the concepts of integral and almost integral.
This property (referred to as the $\Omega$almost integral
property) is a representative independent specialization of the
standard notion of almost integrality. Some of the properties of
this generalization are explored in this paper, and these properties
are compared with the notion of pseudointegrality introduced by
Anderson, Houston, and Zafrullah. Additionally, it is
shown that the $\Omega$almost integral property serves to
characterize the survival/lying over pairs of Dobbs and Coykendall
Keywords:integral closure, complete integral closure Categories:13B22, 13G05, 13B21 

39. CMB 2010 (vol 53 pp. 577)
40. CMB 2010 (vol 53 pp. 667)
 Khashyarmanesh, Kazem

On the Endomorphism Rings of Local Cohomology Modules
Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal
of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also
prove that, for a nonnegative integer $n$ such that
$H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for
all $i >0$ and $z \in \mathfrak{a}$, then
$\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic
image of $R$, where $R_z$ is the ring of fractions of $R$ with
respect to a multiplicatively closed subset $\{z^j \mid j \geqslant
0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z
\in \mathfrak{a}$,
then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$
is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$.
Keywords:local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequence Categories:13D45, 13D07, 13D25 

41. CMB 2010 (vol 53 pp. 602)
42. CMB 2010 (vol 53 pp. 404)
 Broer, Abraham

Invariant Theory of Abelian Transvection Groups
Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called \emph{coregular} if the invariant ring is generated by algebraically independent homogeneous invariants, and the \emph{direct summand property} holds if there is a surjective $k[V]^G$linear map $\pi\colon k[V]\to k[V]^G$. The following ChevalleyShephardTodd type theorem is proved. Suppose $G$ is abelian. Then the action is coregular if and only if $G$ is generated by pseudoreflections and the direct summand property holds.
Category:13A50 

43. CMB 2009 (vol 53 pp. 77)
44. CMB 2009 (vol 53 pp. 247)
 Etingof, P.; Malcolmson, P.; Okoh, F.

Root Extensions and Factorization in Affine Domains
An integral domain R is IDPF (Irreducible Divisors of Powers Finite) if, for every nonzero element a in R, the ascending chain of nonassociate irreducible divisors in R of $a^{n}$ stabilizes on a finite set as n ranges over the positive integers, while R is atomic if every nonzero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension S of R is a \emph{root extension} or \emph{radical extension} if for each s in S, there exists a natural number $n(s)$ with $s^{n(s)}$ in R. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $(R,S)$ is governed by the relative sizes of the unit groups $\operatorname{U}(R)$ and $\operatorname{U}(S)$ and whether S is a root extension of R. The following results are deduced from these considerations. An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let R be a Noetherian domain with integral closure S. Suppose the conductor of S into R is nonzero. Then R is IDPF if and only if S is a root extension of R and $\operatorname{U}(S)/\operatorname{U}(R)$ is finite.
Categories:13F15, 14A25 

45. 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 

46. CMB 2009 (vol 52 pp. 72)
47. CMB 2008 (vol 51 pp. 439)
 Samei, Karim

On the Maximal Spectrum of Semiprimitive Multiplication Modules
An $R$module $M$ is called a multiplication module if for each
submodule $N$ of $M$, $N=IM$ for some ideal $I$ of $R$. As
defined for a commutative ring $R$, an $R$module $M$ is said to be
semiprimitive if the intersection of maximal submodules of $M$ is
zero. The maximal spectra of a semiprimitive multiplication
module $M$ are studied. The isolated points of $\Max(M)$ are
characterized algebraically. The relationships among the maximal
spectra of $M$, $\Soc(M)$ and $\Ass(M)$ are studied. It is shown
that $\Soc(M)$ is exactly the set of all elements of $M$ which
belongs to every maximal submodule of $M$ except for a finite
number. If $\Max(M)$ is infinite, $\Max(M)$ is a onepoint
compactification of a discrete space if and only if $M$ is Gelfand and for
some maximal submodule $K$, $\Soc(M)$ is the intersection of all
prime submodules of $M$ contained in $K$. When $M$ is a
semiprimitive Gelfand module, we prove that every intersection
of essential submodules of $M$ is an essential submodule if and only if
$\Max(M)$ is an almost discrete space. The set of uniform
submodules of $M$ and the set of minimal submodules of $M$
coincide. $\Ann(\Soc(M))M$ is a summand submodule of $M$ if and only if
$\Max(M)$ is the union of two disjoint open subspaces $A$ and
$N$, where $A$ is almost discrete and $N$ is dense in itself. In
particular, $\Ann(\Soc(M))=\Ann(M)$ if and only if $\Max(M)$ is almost
discrete.
Keywords:multiplication module, semiprimitive module, Gelfand module, Zariski topolog Category:13C13 

48. CMB 2008 (vol 51 pp. 406)
 Mimouni, Abdeslam

Condensed and Strongly Condensed Domains
This paper deals with the concepts of condensed and strongly condensed
domains. By definition, an integral domain $R$ is condensed
(resp. strongly condensed) if each pair of ideals $I$ and $J$ of $R$,
$IJ=\{ab/a \in I, b \in J\}$ (resp. $IJ=aJ$ for some $a \in I$ or
$IJ=Ib$ for some $b \in J$). More precisely, we investigate the
ideal theory of condensed and strongly condensed domains in
Noetherianlike settings, especially Mori and strong Mori domains and
the transfer of these concepts to pullbacks.
Categories:13G05, 13A15, 13F05, 13E05 

49. CMB 2007 (vol 50 pp. 598)
50. CMB 2005 (vol 48 pp. 275)
 Smith, Patrick F.

Krull Dimension of Injective Modules Over Commutative Noetherian Rings
Let $R$ be a commutative Noetherian
integral domain with field of fractions $Q$. Generalizing a
fortyyearold theorem of E. Matlis, we prove that the $R$module
$Q/R$ (or $Q$) has Krull dimension if and only if $R$ is semilocal
and onedimensional. Moreover, if $X$ is an injective module over
a commutative Noetherian ring such that $X$ has Krull dimension,
then the Krull dimension of $X$ is at most $1$.
Categories:13E05, 16D50, 16P60 
