Expand all Collapse all | Results 1 - 25 of 55 |
1. CMB Online first
Cover products and Betti polynomials 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 Castelnuovo--Mumford
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:Castelnuovo--Mumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial ideal Categories:13D02, 05E45 |
2. CMB Online first
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 |
3. CMB Online first
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/graph-theoretic 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 |
4. CMB 2014 (vol 58 pp. 134)
On the Generalized Auslander-Reiten Conjecture under Certain Ring Extensions We show under some conditions that a Gorenstein ring $R$ satisfies the
Generalized Auslander-Reiten Conjecture if and only if so does
$R[x]$. When $R$ is a local ring we prove the same result for some
localizations of $R[x]$.
Keywords:Auslander-Reiten conjecture, finitistic extension degree, Gorenstein rings Categories:13D07, 16E30, 16E65 |
5. CMB Online first
On the Generalized Auslander-Reiten Conjecture under Certain Ring Extensions We show under some conditions that a Gorenstein ring $R$ satisfies the
Generalized Auslander-Reiten Conjecture if and only if so does
$R[x]$. When $R$ is a local ring we prove the same result for some
localizations of $R[x]$.
Keywords:Auslander-Reiten conjecture, finitistic extension degree, Gorenstein rings Categories:13D07, 16E30, 16E65 |
6. CMB 2014 (vol 57 pp. 573)
Some Results on the Domination Number of a Zero-divisor Graph In this paper, we investigate the domination, total domination and
semi-total domination numbers of a zero-divisor 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:zero-divisor graph, domination number Categories:05C75, 13H10 |
7. CMB 2014 (vol 57 pp. 477)
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 long-standing problem to find
out the cases giving equality. Thanks to the formal grade we give
conditions in which the above inequalities become
equalities.
Keywords:set-theoretically and cohomologically complete intersection ideals, analytic spread, monomials, formal grade, depth of powers of ideals Categories:13D45, 13C14 |
8. CMB 2014 (vol 57 pp. 735)
On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras |
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
non-zero elements $\alpha,\beta\in F$?
Keywords:uniserial module, Lie algebra, associative algebra, primitive element Categories:17B10, 13C05, 12F10, 12E20 |
9. 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 |
10. CMB 2013 (vol 57 pp. 188)
A Characterization of Bipartite Zero-divisor Graphs In this paper we obtain a characterization for all bipartite
zero-divisor graphs of commutative rings $R$ with $1$, such that
$R$ is finite or $|Nil(R)|\neq2$.
Keywords:zero-divisor graph, bipartite graph Categories:13AXX, 05C25 |
11. CMB 2013 (vol 57 pp. 159)
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 0-dimensional
module. It is also an extension of strongly 0-dimensional rings. After
this we investigate properties of strongly 0-dimensional modules and give
relations of von Neumann regular modules, Q-modules and strongly
0-dimensional modules.
Keywords:strongly 0-dimensional rings, Q-module, Von Neumann regular module Categories:13C99, 16D10 |
12. CMB 2012 (vol 57 pp. 289)
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 non-negative 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 |
13. 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 |
14. CMB 2012 (vol 56 pp. 551)
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
countable-dimensional 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 |
15. CMB 2012 (vol 56 pp. 534)
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 Kaniuth-Lau-Pym
and the second named author in the special case that $\pi \colon A
\longrightarrow \mathbf C$ is a non-zero character on $A$.
Keywords:Banach algebras, $\pi$-invariance, derivations, representations Categories:46H15, 46H25, 13N15 |
16. CMB 2012 (vol 56 pp. 491)
A Note on Homological Dimensions of Artinian Local Cohomology Modules Let $(R,{\frak m})$ be a non-zero commutative Noetherian local ring
(with identity), $M$ be a non-zero 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 |
17. CMB 2011 (vol 56 pp. 407)
On Domination in Zero-Divisor Graphs We first determine the domination number for the zero-divisor
graph of the product of two commutative rings with $1$. We then
calculate the domination number for the zero-divisor graph of any
commutative artinian ring. Finally, we extend some of the results
to non-commutative rings in which an element is a left
zero-divisor if and only if it is a right zero-divisor.
Keywords:zero-divisor graph, domination Categories:13AXX, 05C69 |
18. CMB 2011 (vol 55 pp. 752)
Approximation of Holomorphic Solutions of a System of Real Analytic Equations We prove the existence of an approximation function for holomorphic
solutions of a system of real analytic equations. For this we use
ultraproducts and Weierstrass systems introduced by J. Denef and L.
Lipshitz. We also prove a version of the PÅoski smoothing theorem in
this case.
Keywords:Artin approximation, real analytic equations Categories:13B40, 13L05, 14F12 |
19. CMB 2011 (vol 56 pp. 31)
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 |
20. 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 |
21. 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 |
22. CMB 2011 (vol 55 pp. 315)
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 well-known 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 ring-theoretic vanishing criterion for local cohomology modules.
Category:13D45 |
23. CMB 2011 (vol 55 pp. 81)
Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals Let $\mathfrak a$ be an ideal of a commutative Noetherian
ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our
main result asserts that if $\dim R/\mathfrak a\leq 1$, then all generalized
local cohomology modules $H^i_{\mathfrak a}(M,N)$ are $\mathfrak a$-cofinite.
Keywords:cofinite modules, generalized local cohomology modules, local cohomology modules Categories:13D45, 13E05, 13E10 |
24. CMB 2011 (vol 55 pp. 153)
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 |
25. CMB 2011 (vol 54 pp. 619)
Artinian and Non-Artinian 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 Cohen-Macaulay 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
non-empty 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 |