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Search: All articles in the CMB digital archive with keyword ideal

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1. CMB 2015 (vol 58 pp. 320)

Llamas, Aurora; Martínez-Bernal, 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 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 idealCategories:13D02, 05E45

2. CMB 2015 (vol 58 pp. 350)

Merino-Cruz, 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 idealCategories:46J15, 46J20, 30H50

3. 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 depthCategories:13F20, 13C15

4. 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 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 idealsCategories:13D45, 13C14

5. 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 non-compactness 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 non-compactness, condensing map, partially additive operator, regular space, ideal spaceCategories:47H08, 46E30, 47H99, 47G10

6. 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 regularizationCategory:46L05

7. 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 Stone--Äech compactification of $G$. We take the points of $Äta G$ 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 two-sided 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:Stone--Äech compactification, uniform ultrafilter, closed left ideal, decompositionCategories:22A15, 54H20, 22A30, 54D80

8. 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 propertyCategories:46A40, 46B42, 46B45

9. CMB 2011 (vol 54 pp. 487)

Kong, Xiangjun
 Some Properties Associated with Adequate Transversals In this paper, another relationship between the quasi-ideal 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 quasi-ideal 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, quasi-idealCategory:20M10

10. 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 spacesCategories:46M05, 46M35, 46A32

11. CMB 2010 (vol 53 pp. 577)

 A Unified Approach to Local Cohomology Modules using Serre Classes This paper discusses the connection between the local cohomology modules and the Serre classes of $R$-modules. This connection has provided a common language for expressing some results regarding the local cohomology $R$-modules that have appeared in different papers. Keywords:associated prime ideals, local cohomology modules, Serre classCategory:13D45

12. CMB 2005 (vol 48 pp. 121)

Mollin, R. A.
 Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D=2^hc$ where $c>1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be $2^h$. At the end of the paper, we also address the case where $D=c$ is odd and the central norm of $\sqrt{D}$ is equal to $2$. Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fieldsCategories:11A55, 11D09, 11R11

13. 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 identityCategories:46H20, 46H10, 46H25

14. 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 weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property. Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions, $M$-idealsCategories:46B20, 46E40

15. 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

16. 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 idealsCategories:20C07, 16A27
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