Expand all Collapse all | Results 1 - 4 of 4 |
1. CMB 2013 (vol 57 pp. 413)
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) 124-127], 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 non-unit 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 |
2. CMB 2012 (vol 56 pp. 683)
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)
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 $M$ H-smaller (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 non-Noetherian 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. 127)
Characterizations of Three Classes of Zero-Divisor Graphs The zero-divisor graph $\Gamma(R)$ of a commutative ring $R$ is the graph whose vertices consist of
the nonzero zero-divisors 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 zero-divisor graphs of Boolean
rings. Also, commutative rings $R$ such that
$\Gamma(R)$ is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as
those whose zero-divisor graphs are central vertex complete.
Categories:13A99, 13M99 |