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Search: MSC category 13 ( Commutative rings and algebras )

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1. CMB Online first

Boynton, Jason Greene; Coykendall, Jim
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

2. CMB Online first

Nasseh, Saeed
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

3. CMB Online first

Nasseh, Saeed
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

4. CMB 2014 (vol 57 pp. 573)

Kiani, Sima; Maimani, Hamid Reza; Nikandish, Reza
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

5. 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 ideals
Categories:13D45, 13C14

6. CMB 2014 (vol 57 pp. 735)

Cagliero, Leandro; Szechtman, Fernando
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

7. CMB 2013 (vol 57 pp. 413)

Samei, Karim
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

8. CMB 2013 (vol 57 pp. 188)

Rad, Nader Jafari; Jafari, Sayyed Heidar
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

9. CMB 2013 (vol 57 pp. 159)

Oral, Kürşat Hakan; Özkirişci, Neslihan Ayşen; Tekir, Ünsal
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

10. CMB 2012 (vol 57 pp. 289)

Ghasemi, Mehdi; Marshall, Murray; Wagner, Sven
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

11. 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

12. CMB 2012 (vol 56 pp. 551)

Handelman, David
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

13. CMB 2012 (vol 56 pp. 534)

Filali, M.; Monfared, M. Sangani
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

14. CMB 2012 (vol 56 pp. 491)

Bahmanpour, Kamal
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

15. CMB 2011 (vol 56 pp. 407)

Rad, Nader Jafari; Jafari, Sayyed Heidar; Mojdeh, Doost Ali
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

16. CMB 2011 (vol 55 pp. 752)

Hickel, M.; Rond, G.
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 Płoski smoothing theorem in this case.

Keywords:Artin approximation, real analytic equations
Categories:13B40, 13L05, 14F12

17. 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

18. 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 $M$ 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 Jónsson modules.

Keywords:Noetherian ring, residually finite ring, cardinal number, continuum hypothesis, valuation ring, Jónsson module
Categories:13A99, 13C05, 13E05, 03E50

19. CMB 2011 (vol 55 pp. 127)

LaGrange, John D.
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

20. 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 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

21. CMB 2011 (vol 55 pp. 81)

Divaani-Aazar, Kamran; Hajikarimi, Alireza
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

22. 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

23. CMB 2011 (vol 54 pp. 619)

Dibaei, Mohammad T.; Vahidi, Alireza
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

24. CMB 2011 (vol 54 pp. 716)

Okassa, Eugène
Symplectic Lie-Rinehart-Jacobi Algebras and Contact Manifolds
We give a characterization of contact manifolds in terms of symplectic Lie-Rinehart-Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold.

Keywords:Lie-Rinehart algebras, differential operators, Jacobi manifolds, symplectic manifolds, contact manifolds
Categories:13N05, 53D05, 53D10

25. 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, Kähler 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
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