The Zariski topology-graph of modules over commutative rings II

Abstract Let M be a module over a commutative ring R. In this paper, we continue our study about the Zariski topology-graph $$G(\tau _T)$$ G ( τ T ) which was introduced in Ansari-Toroghy et al. (Commun Algebra 42:3283–3296, 2014). For a non-empty subset T of $$\mathrm{Spec}(M)$$ Spec ( M ) , we obtain useful characterizations for those modules M for which $$G(\tau _T)$$ G ( τ T ) is a bipartite graph. Also, we prove that if $$G(\tau _T)$$ G ( τ T ) is a tree, then $$G(\tau _T)$$ G ( τ T ) is a star graph. Moreover, we study coloring of Zariski topology-graphs and investigate the interplay between $$\chi (G(\tau _T))$$ χ ( G ( τ T ) ) and $$\omega (G(\tau _T))$$ ω ( G ( τ T ) ) .

On the graph of modules over commutative rings II

Filomat ◽

10.2298/fil1810657a ◽

2018 ◽

Vol 32 (10) ◽

pp. 3657-3665

Author(s):

Habibollah Ansari-Toroghy ◽

Shokoufeh Habibi ◽

Masoomeh Hezarjaribi

Keyword(s):

Commutative Ring ◽

Bipartite Graph ◽

Rocky Mountain ◽

Commutative Rings ◽

Star Graph ◽

Zariski Topology ◽

Empty Subset ◽

Let M be a module over a commutative ring R. In this paper, we continue our study about the quasi-Zariski topology-graph G(?*T) which was introduced in (On the graph of modules over commutative rings, Rocky Mountain J. Math. 46(3) (2016), 1-19). For a non-empty subset T of Spec(M), we obtain useful characterizations for those modules M for which G(?*T) is a bipartite graph. Also, we prove that if G(?*T) is a tree, then G(?*T) is a star graph. Moreover, we study coloring of quasi-Zariski topology-graphs and investigate the interplay between ?(G(?+T)) and ?(G(?+T)).

The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2018-0032 ◽

2018 ◽

Vol 26 (3) ◽

pp. 41-56

Author(s):

H. Ansari-Toroghy ◽

Sh. Habibi

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Zariski Topology ◽

Topological Properties ◽

Maximal Spectrum ◽

AbstractLet M be a module over a commutative ring and let Max(M) be the collection of all maximal submodules of M. We topologize Max(M) with quasi-Zariski topology, where M is a Max-top module. For a subset T of Max(M), we introduce a new graph $G(\tau_T^{*m})$, called the quasi-Zariski topology-graph on the maximal spectrum of M. It helps us to study algebraic (resp. topological) properties of M (resp. Max(M)) by using the graphs theoretical tools.

CLEAN, ALMOST CLEAN, POTENT COMMUTATIVE RINGS

10.1142/s0219498807002466 ◽

2007 ◽

Vol 06 (04) ◽

pp. 671-685 ◽

Cited By ~ 2

Author(s):

K. VARADARAJAN

Keyword(s):

Commutative Ring ◽

Analogous Result ◽

Commutative Rings ◽

Complete Characterization ◽

Zariski Topology ◽

Clean Ring ◽

Regular Rings

We give a complete characterization of the class of commutative rings R possessing the property that Spec(R) is weakly 0-dimensional. They turn out to be the same as strongly π-regular rings. We considerably strengthen the results of K. Samei [13] tying up cleanness of R with the zero dimensionality of Max(R) in the Zariski topology. In the class of rings C(X), W. Wm Mc Govern [6] has characterized potent rings as the ones with X admitting a clopen π-base. We prove the analogous result for any commutative ring in terms of the Zariski topology on Max(R). Mc Govern also introduced the concept of an almost clean ring and proved that C(X) is almost clean if and only if it is clean. We prove a similar result for all Gelfand rings R with J(R) = 0.

The Logical Complexity of Finitely Generated Commutative Rings

International Mathematics Research Notices ◽

10.1093/imrn/rny023 ◽

2018 ◽

Vol 2020 (1) ◽

pp. 112-166 ◽

Cited By ~ 1

Author(s):

Matthias Aschenbrenner ◽

Anatole Khélif ◽

Eudes Naziazeno ◽

Thomas Scanlon

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Prime Ideals ◽

Zariski Topology ◽

Dual Numbers ◽

Finitely Generated ◽

Nonstandard Model

AbstractWe characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with $(\mathbb{N},{+},{\times })$ if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in $\mathbb{Z}$. Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers over $\mathbb{Z}$ is not bi-interpretable with $\mathbb{N}$.

The Zariski Topology-Graph of Modules Over Commutative Rings

Communications in Algebra ◽

10.1080/00927872.2013.780065 ◽

2014 ◽

Vol 42 (8) ◽

pp. 3283-3296 ◽

Cited By ~ 3

Author(s):

H. Ansari-Toroghy ◽

Sh. Habibi

Keyword(s):

Commutative Rings ◽

Zariski Topology ◽

Zariski Topology

Formalized Mathematics ◽

10.2478/forma-2018-0024 ◽

2018 ◽

Vol 26 (4) ◽

pp. 277-283

Author(s):

Yasushige Watase

Keyword(s):

Commutative Ring ◽

Local Ring ◽

Jacobson Radical ◽

Commutative Rings ◽

Ring Homomorphism ◽

Ring Theory ◽

Prime Ideals ◽

Topological Spaces ◽

Zariski Topology ◽

Prime Spectrum

Summary We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1]. The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : A → B, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h−1(𝔭) where 𝔭 2 Spec B.

N-ideals of commutative rings

Filomat ◽

10.2298/fil1710933t ◽

2017 ◽

Vol 31 (10) ◽

pp. 2933-2941 ◽

Cited By ~ 3

Author(s):

Unsal Tekir ◽

Suat Koc ◽

Kursat Oral

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Proper Ideal ◽

Prime Ideals ◽

Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

On commutative rings with uniserial dimension

10.1142/s0219498815500085 ◽

2014 ◽

Vol 14 (01) ◽

pp. 1550008 ◽

Cited By ~ 2

Author(s):

A. Ghorbani ◽

Z. Nazemian

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Artinian Rings ◽

Uniform Dimension ◽

Uniserial Modules

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

10.1142/s021949881250199x ◽

2013 ◽

Vol 12 (04) ◽

pp. 1250199 ◽

Cited By ~ 11

Author(s):

T. ASIR ◽

T. TAMIZH CHELVAM

Keyword(s):

Commutative Ring ◽

Independence Number ◽

Clique Number ◽

Commutative Rings ◽

Intersection Graph ◽

Vertex Connectivity ◽

Total Graph ◽

Graph Theoretic ◽

Vertex Set ◽

Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

ON RINGS R WHOSE GRAPHS Γ(R) SATISFY CONDITION (Kp)

10.1142/s0219498811004835 ◽

2011 ◽

Vol 10 (04) ◽

pp. 665-674

Author(s):

LI CHEN ◽

TONGSUO WU

Keyword(s):

Prime Number ◽

Commutative Ring ◽

Complete Graph ◽

Zero Divisor ◽

Commutative Rings ◽

Satisfy Condition ◽

Induced Subgraph ◽

Zero Divisor Graph ◽

Ring Graph ◽

Finite Commutative Rings

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).