scholarly journals THE REGULAR GRAPH OF A NONCOMMUTATIVE RING

2013 ◽  
Vol 89 (1) ◽  
pp. 132-140 ◽  
Author(s):  
S. AKBARI ◽  
F. HEYDARI
Keyword(s):  
Regular Graph ◽  
Noetherian Ring ◽  
Clique Number ◽  
Prime Ideals ◽  
Induced Subgraph ◽  
Total Graph ◽  
Arbitrary Ring ◽  
Regular Elements ◽  
Zero Divisors ◽  
Minimal Prime

AbstractLet $R$ be a ring and $Z(R)$ be the set of all zero-divisors of $R$. The total graph of $R$, denoted by $T(\Gamma (R))$ is a graph with all elements of $R$ as vertices, and two distinct vertices $x, y\in R$ are adjacent if and only if $x+ y\in Z(R)$. Let the regular graph of $R$, $\mathrm{Reg} (\Gamma (R))$, be the induced subgraph of $T(\Gamma (R))$ on the regular elements of $R$. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set $\{ 3, 4, \infty \} $. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if $R$ is a reduced left Noetherian ring and $2\not\in Z(R)$, then the chromatic number and the clique number of $\mathrm{Reg} (\Gamma (R))$ are the same and they are ${2}^{r} $, where $r$ is the number of minimal prime ideals of $R$. Among other results, we show that if $R$ is a semiprime left Noetherian ring and $\mathrm{Reg} (R)$ is finite, then $R$ is finite.

The M-Regular Graph of a Commutative Ring

2015 ◽  
Vol 65 (1) ◽  
pp. 1-12
Author(s):  
M. J. Nikmehr ◽  
F. Heydari
Keyword(s):  
Commutative Ring ◽  
Lower Bounds ◽  
Regular Graph ◽  
Noetherian Ring ◽  
Undirected Graph ◽  
Independence Number ◽  
Clique Number ◽  
Finitely Generated Module ◽  
Regular Elements ◽  
Zero Divisors

AbstractLet R be a commutative ring and M be an R-module, and let Z(M) be the set of all zero-divisors on M. In 2008, D. F. Anderson and A. Badawi introduced the regular graph of R. In this paper, we generalize the regular graph of R to the M-regular graph of R, denoted by M-Reg(Γ(R)). It is the undirected graph with all M-regular elements of R as vertices, and two distinct vertices x and y are adjacent if and only if x+y ∈ Z(M). The basic properties and possible structures of M-Reg(Γ(R)) are studied. We determine the girth of the M-regular graph of R. Also, we provide some lower bounds for the independence number and the clique number of M- Reg(Γ(R)). Among other results, we prove that for every Noetherian ring R and every finitely generated module M over R, if 2 ∉ Z(M) and the independence number of M-Reg(Γ(R)) is finite, then R is finite.

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

Semicritical Rings and the Quotient Problem

1984 ◽  
Vol 27 (2) ◽  
pp. 160-170
Author(s):  
Karl A. Kosler
Keyword(s):  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Regular Elements ◽  
The Right ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Quotient Rings ◽  
The Relationship

AbstractThe purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

scholarly journals Thek-Zero-Divisor Hypergraph of a Commutative Ring

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Ch. Eslahchi ◽  
A. M. Rahimi
Keyword(s):  
Commutative Ring ◽  
Nilpotent Element ◽  
Zero Divisor ◽  
Clique Number ◽  
Common Point ◽  
Prime Ideals ◽  
Uniform Hypergraph ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and thek-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. LetRbe a commutative ring andkan integer strictly larger than2. Ak-uniform hypergraphHk(R)with the vertex setZ(R,k), the set of allk-zero-divisors inR, is associated toR, where eachk-subset ofZ(R,k)that satisfies thek-zero-divisor condition is an edge inHk(R). It is shown that ifRhas two prime idealsP1andP2with zero their only common point, thenHk(R)is a bipartite (2-colorable) hypergraph with partition setsP1−Z′andP2−Z′, whereZ′is the set of all zero divisors ofRwhich are notk-zero-divisors inR. IfRhas a nonzero nilpotent element, then a lower bound for the clique number ofH3(R)is found. Also, we have shown thatH3(R)is connected with diameter at most 4 wheneverx2≠0for all3-zero-divisorsxofR. Finally, it is shown that for any finite nonlocal ringR, the hypergraphH3(R)is complete if and only ifRis isomorphic toZ2×Z2×Z2.

Coloring of lattices

2010 ◽  
Vol 60 (4) ◽  
Author(s):  
S. Nimbhokar ◽  
M. Wasadikar ◽  
M. Pawar
Keyword(s):  
Distributive Lattice ◽  
Chromatic Number ◽  
Finite Lattice ◽  
Clique Number ◽  
Prime Ideals ◽  
Minimal Prime

AbstractThe concept of coloring is studied for graphs derived from lattices with 0. It is shown that, if such a graph is derived from an atomic or distributive lattice, then the chromatic number equals the clique number. If this number is finite, then in the case of a distributive lattice, it is determined by the number of minimal prime ideals in the lattice. An estimate for the number of edges in such a graph of a finite lattice is given.

The spectrum subgraph of the annihilating-ideal graph of a commutative ring

2015 ◽  
Vol 14 (08) ◽  
pp. 1550130
Author(s):  
R. Taheri ◽  
M. Behboodi ◽  
A. Tehranian
Keyword(s):  
Commutative Ring ◽  
Complete Graph ◽  
Connected Graph ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Star Graph ◽  
Induced Subgraph ◽  
Artinian Rings ◽  
Spectrum Graph

In this paper we introduce and study the spectrum graph of a commutative ring R, denoted by 𝔸𝔾s(R), that is, the graph whose vertices are all non-zero prime ideals of R with non-zero annihilator and two distinct vertices P1, P2 are adjacent if and only if P1P2 = (0). This is an induced subgraph of the annihilating-ideal graph 𝔸𝔾(R) of R. Among other results, we present the structures of all graphs which can be realized as the spectrum graph of a commutative ring. Then we show that for a non-domain Noetherian ring R, 𝔸𝔾s(R), is a connected graph if and only if 𝔸𝔾s(R) is a star graph if and only if 𝔸𝔾s(R) ≅ K1, K2 or K1,∞, where Kn is a complete graph with n vertices and K1,∞ is a star graph with infinite vertices. Also, we completely characterize the spectrum graphs of Artinian rings. Finally, as an application, we present some relationships between the annihilating-ideal graph and its spectrum subgraph.

scholarly journals On the Diameter of Dual Graphs of Stanley-Reisner Rings and Hirsch Type Bounds on Abstractions of Polytopes

10.37236/6831 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Brent Holmes
Keyword(s):  
Noetherian Ring ◽  
Dual Graph ◽  
Upper Bounds ◽  
Maximum Diameter ◽  
Prime Ideals ◽  
Lower And Upper Bounds ◽  
Commutative Noetherian Ring ◽  
Positive Dimension ◽  
Dual Graphs ◽  
Minimal Prime

Let $R$ be an equidimensional commutative Noetherian ring of positive dimension. The dual graph $\mathcal{G} (R)$ of $R$ is defined as follows: the vertices are the minimal prime ideals of $R$, and the edges are the pairs of prime ideals $(P_1,P_2)$ with height$(P_1 + P_2) = 1$. If $R$ satisfies Serre's property $(S_2)$, then $\mathcal{G} (R)$ is connected. In this note, we provide lower and upper bounds for the maximum diameter of dual graphs of Stanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number of variables and the dimension. Dual graphs of $(S_2)$ Stanley-Reisner rings are a natural abstraction of the $1$-skeletons of polyhedra. We discuss how our bounds imply new Hirsch-type bounds on $1$-skeletons of polyhedra.

On Maximal Torsion Radicals

1973 ◽  
Vol 25 (4) ◽  
pp. 712-726 ◽  
Author(s):  
John A. Beachy
Keyword(s):  
General Result ◽  
Noetherian Ring ◽  
Associative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Maximal Torsion ◽  
Minimal Prime

Let R be an associative ring with identity, and let denote the category of unital left R-modules. It is known that if R is a commutative, Noetherian ring, then the maximal torsion radicals of correspond to the minimal prime ideals of R. In fact, Nӑstӑsescu and Popescu [15, Proposition 2.7] have given a more general result valid for arbitrary commutative rings. This paper investigates maximal torsion radicals over rings not necessarily commutative.

TOTAL GRAPHS OF POLYNOMIAL RINGS AND RINGS OF FRACTIONS

2013 ◽  
Vol 05 (04) ◽  
pp. 1350035
Author(s):  
MOJGAN AFKHAMI ◽  
KAZEM KHASHYARMANESH
Keyword(s):  
Commutative Ring ◽  
Independence Number ◽  
Clique Number ◽  
Polynomial Rings ◽  
Chromatic Index ◽  
Total Graph ◽  
Zero Divisors

Let R be a commutative ring. The total graph of R, denoted by T(Γ(R)), is a graph with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x + y ∈ Z(R), where Z(R) denotes the set of zero-divisors of R. In this paper, we examine the preservation of the diameter, girth and completeness of T(Γ(R)) under extension to polynomial rings and rings of fractions. We also study the chromatic index, clique number and independence number of T(Γ(R)).

scholarly journals On two-sided artinian quotient rings

1972 ◽  
Vol 13 (2) ◽  
pp. 159-163 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Regular Element ◽  
Quotient Ring ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Zero Divisors ◽  
Quotient Rings

Djabali [1] has proved that, if R is a right and left noetherian ring with an identity and if the proper prime ideals of R are maximal, then R has a right and left artinian two-sided quotient ring. Robson [5, Theorem 2.11] and Small [6, Theorem 2.13] have proved independently that, if Ris a commutative noetherian ring, then Rhas an artinian quotient ring if and only if the prime ideals of Rthat belong to the zero ideal are all minimal. We shall generalise these results by proving theTheorem. Let R be a right and left noetherian ring with a regular element. Then R has a right and left artinian two-sided quotient ring if and only if each prime ideal of R consisting of zero–divisors is minimal.

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