1. 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 

2. 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 

3. 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 

4. CMB 2011 (vol 55 pp. 81)
5. 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 

6. 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 
