scholarly journals THE ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS I

2011 ◽  
Vol 10 (04) ◽  
pp. 727-739 ◽  
Author(s):  
M. BEHBOODI ◽  
Z. RAKEEI
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Connected Graph ◽  
Undirected Graph ◽  
Chain Condition ◽  
Commutative Rings ◽  
Finiteness Conditions ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

Let R be a commutative ring, with 𝔸(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). It is the (undirected) graph with vertices 𝔸(R)* ≔ 𝔸(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of 𝔸𝔾(R). For instance, it is shown that if R is not a domain, then 𝔸𝔾(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of 𝔸𝔾(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, 𝔸𝔾(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of 𝔸𝔾(R). It is shown that 𝔸𝔾(R) is a connected graph and diam (𝔸𝔾)(R) ≤ 3 and if 𝔸𝔾(R) contains a cycle, then gr (𝔸𝔾(R)) ≤ 4. Also, rings R for which the graph 𝔸𝔾(R) is complete or star, are characterized, as well as rings R for which every vertex of 𝔸𝔾(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

CO-MAXIMAL IDEAL GRAPHS OF COMMUTATIVE RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250114 ◽  
Author(s):  
MENG YE ◽  
TONGSUO WU
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Maximal Ideal ◽  
Connected Graph ◽  
Jacobson Radical ◽  
Clique Number ◽  
Commutative Rings ◽  
Maximal Ideals ◽  
Simple Connected Graph

In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use [Formula: see text] to denote this graph, with its vertices the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. We show some properties of this graph. For example, this graph is a simple, connected graph with diameter less than or equal to three, and both the clique number and the chromatic number of the graph are equal to the number of maximal ideals of the ring R.

scholarly journals Commutative rings whose quotients are Goldie

1975 ◽  
Vol 16 (1) ◽  
pp. 32-33 ◽  
Author(s):  
Victor P. Camillo
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Chain Condition ◽  
Commutative Rings ◽  
The Other ◽  
Direct Sums ◽  
Other Hand ◽  
Commutative Domain ◽  
Ascending Chain Condition

All rings considered here have units. A (non-commutative) ring is right Goldieif it has no infinite direct sums of right ideals and has the ascending chain condition on annihilator right ideals. A right ideal A is an annihilator if it is of the form {a ∈ R/xa = 0 for all x ∈ X}, where X is some subset of R. Naturally, any noetherian ring is Goldie, but so is any commutative domain, so that the converse is not true. On the other hand, since any quotient ring of a noetherian ring is noetherian, it is true that every quotient is Goldie. A reasonable question therefore is the following: must a ring, such that every quotient ring is Goldie, be noetherian? We prove the following theorem:Theorem. A commutative ring is noetherian if and only if every quotient is Goldie.

scholarly journals Maximal Ideal Graph of Commutative Rings

2020 ◽  
pp. 2070-2076
Author(s):  
F. H. Abdulqadr
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Vertex Set

In this paper, we introduce and study the notion of the maximal ideal graph of a commutative ring with identity. Let R be a commutative ring with identity. The maximal ideal graph of R, denoted by MG(R), is the undirected graph with vertex set, the set of non-trivial ideals of R, where two vertices I1 and I2 are adjacent if I1 I2 and I1+I2 is a maximal ideal of R. We explore some of the properties and characterizations of the graph.

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

2020 ◽  
Vol 27 (03) ◽  
pp. 531-544
Author(s):  
Farid Kourki ◽  
Rachid Tribak
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Chain Condition ◽  
Finitely Generated ◽  
Artinian Modules ◽  
Cyclic Submodules ◽  
Descending Chain Condition

A module satisfying the descending chain condition on cyclic submodules is called coperfect. The class of coperfect modules lies properly between the class of locally artinian modules and the class of semiartinian modules. Let R be a commutative ring with identity. We show that every semiartinian R-module is coperfect if and only if R is a T-ring. It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if 𝔪/𝔪2 is a finitely generated R-module for every maximal ideal 𝔪 of R.

The flat topology on the minimal and maximal prime spectrum of a commutative ring

2019 ◽  
Vol 18 (06) ◽  
pp. 1950110
Author(s):  
Esmaeil Rostami ◽  
Masoumeh Hedayati ◽  
Nosratollah Shajareh Poursalavati
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Chain Condition ◽  
Commutative Rings ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Compact Topological Space ◽  
Algebraic Properties ◽  
Ascending Chain Condition

In this paper, we investigate connections between some algebraic properties of commutative rings and topological properties of their minimal and maximal prime spectrum with respect to the flat topology. We show that for a commutative ring [Formula: see text], the ascending chain condition on principal annihilator ideals of [Formula: see text] holds if and only if [Formula: see text] is a Noetherian topological space as a subspace of [Formula: see text] with respect to the flat topology and we give a characterization for a topological space [Formula: see text] for which [Formula: see text] is a Noetherian topological space as a subspace of [Formula: see text] with respect to the flat topology. Also, we give a characterization for rings whose maximal prime spectrum is a compact topological space with respect to the flat topology. Some other results are obtained too.

Completely Indecomposable Modules

1949 ◽  
Vol 1 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Chain Condition ◽  
Indecomposable Module ◽  
Unit Element ◽  
Indecomposable Modules ◽  
Operator Domain ◽  
Unit Operator ◽  
Ascending Chain Condition

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

Bounds for the Genus of Generalized Total Graph of a Commutative Ring

2019 ◽  
Vol 26 (03) ◽  
pp. 519-528
Author(s):  
T. Asir ◽  
K. Mano
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Total Graph

Let R be a commutative ring with non-zero identity and I its proper ideal. The total graph of R with respect to I, denoted by T (ΓI (R)), is the undirected graph with all elements of R as vertices, and where distinct vertices x and y are adjacent if and only if [Formula: see text]. In this paper, some bounds for the genus of T(ΓI(R)) are obtained. We improve and generalize some results for the total graphs of commutative rings. In addition, we obtain an isomorphism relation between two ideal based total graphs.

Varieties generated by 2-testable monoids

2012 ◽  
Vol 49 (3) ◽  
pp. 366-389 ◽  
Author(s):  
Edmond Lee
Keyword(s):  
Present Article ◽  
Chain Condition ◽  
Finite Basis ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals

1971 ◽  
Vol 14 (3) ◽  
pp. 443-444 ◽  
Author(s):  
Kwangil Koh ◽  
A. C. Mewborn
Keyword(s):  
Prime Ring ◽  
Chain Condition ◽  
Prime Rings ◽  
Simple Ring ◽  
Descending Chain Condition ◽  
Image Position ◽  
The Right ◽  
Ascending Chain Condition

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such thatI is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

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