On projective intersection graph of ideals of commutative rings

2019 ◽  
pp. 2150017
Author(s):  
V. Ramanathan
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Principal Ideal ◽  
Artinian Ring ◽  
Intersection Graph ◽  
Principal Ideal Ring ◽  
Ideal Formula ◽  
Ideal Ring ◽  
Projective Graph ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] the set of all nontrivial proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as the set [Formula: see text], and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] where [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] of nilpotency three and [Formula: see text] is a field. Furthermore, it is shown that for an Artinian ring [Formula: see text] [Formula: see text] if and only if [Formula: see text] where each [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] such that [Formula: see text]

Commutative rings whose proper ideals are direct sum of completely cyclic modules

2016 ◽  
Vol 15 (09) ◽  
pp. 1650160 ◽  
Author(s):  
M. Behboodi ◽  
S. Heidari ◽  
S. Roointan-Isfahani
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Principal Ideal ◽  
Simple Formula ◽  
Commutative Rings ◽  
Index Set ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring

By two results of Köthe and Cohen–Kaplansky we obtain that “a commutative ring [Formula: see text] has the property that every [Formula: see text]-module is a direct sum of (completely) cyclic modules if and only if [Formula: see text] is an Artinian principal ideal ring” (an [Formula: see text]-module [Formula: see text] is called completely cyclic if each submodule of [Formula: see text] is cyclic). In this paper, we describe and study commutative rings whose proper ideals are direct sum of completely cyclic modules. It is shown that every proper ideal of a commutative ring [Formula: see text] is a direct sum of completely cyclic [Formula: see text]-modules if and only if [Formula: see text] is a principal ideal ring or [Formula: see text] is a local ring with maximal ideal [Formula: see text] such that there is an index set [Formula: see text] and a set of elements [Formula: see text] such that [Formula: see text] with each [Formula: see text] a simple [Formula: see text]-module and [Formula: see text] a principal ideal ring.

Special Principal Ideal Rings and Absolute Subretracts

1991 ◽  
Vol 34 (3) ◽  
pp. 364-367 ◽  
Author(s):  
Eric Jespers
Keyword(s):  
Maximal Ideal ◽  
Principal Ideal ◽  
Identity Mapping ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Theoretic Version

AbstractA ring R is said to be an absolute subretract if for any ring S in the variety generated by R and for any ring monomorphism f from R into S, there exists a ring morphism g from S to R such that gf is the identity mapping. This concept, introduced by Gardner and Stewart, is a ring theoretic version of an injective notion in certain varieties investigated by Davey and Kovacs.Also recall that a special principal ideal ring is a local principal ring with nonzero nilpotent maximal ideal. In this paper (finite) special principal ideal rings that are absolute subretracts are studied.

When is every module with essential socle a direct sum of automorphism-invariant modules?

2019 ◽  
Vol 19 (10) ◽  
pp. 2050185
Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Saboura Dolati Pish Hesari ◽  
Mehdi Khoramdel
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Essential Extension ◽  
Principal Ideal Ring ◽  
Von Neumann ◽  
Simple Modules ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Ideal Ring ◽  
Von Neumann Regular

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if [Formula: see text] is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then [Formula: see text] is right Noetherian. Also, we show a von Neumann regular (semiregular) ring [Formula: see text] with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

Rings Whose Modules are ⊕ -Cofinitely Supplemented

2013 ◽  
Vol 0 (0) ◽  
Author(s):  
Ergül Türkmen
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Proper Generalization

Abstract It is known that a commutative ring R is an artinian principal ideal ring if and only if every left R-module is ⊕-supplemented. In this paper, we show that a commutative ring R is a semiperfect principal ideal ring if every left R-module is ⊕-cofinitely supplemented. The converse holds if R is a max ring. Moreover, we study maximally ⊕- supplemented modules as a proper generalization of ⊕-cofinitely supplemented modules. Using these modules, we also prove that a ring R is semiperfect if and only if every projective left R-module with small radical is supplemented.

scholarly journals On some properties of⊕-supplemented modules

2003 ◽  
Vol 2003 (69) ◽  
pp. 4373-4387 ◽  
Author(s):  
A. Idelhadj ◽  
R. Tribak
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Principal Ideal ◽  
Finitely Generated ◽  
Principal Ideal Ring ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Ideal Ring

A moduleMis⊕-supplemented if every submodule ofMhas a supplement which is a direct summand ofM. In this paper, we show that a quotient of a⊕-supplemented module is not in general⊕-supplemented. We prove that over a commutative ringR, every finitely generated⊕-supplementedR-moduleMhaving dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ringRis semisimple if and only if the class of⊕-supplementedR-modules coincides with the class of injectiveR-modules. The structure of⊕-supplemented modules over a commutative principal ideal ring is completely determined.

Polynomials inducing the zero function on chain rings

2018 ◽  
Vol 17 (08) ◽  
pp. 1850160 ◽  
Author(s):  
Mark W. Rogers ◽  
Cameron Wickham
Keyword(s):  
Maximal Ideal ◽  
Principal Ideal ◽  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Minimal Set ◽  
Principal Ideal Ring ◽  
Ideal Formula ◽  
Zero Function ◽  
The Ideal

We provide a minimal set of generators for the ideal of polynomials in [Formula: see text] that map the maximal ideal [Formula: see text] into one of its powers [Formula: see text], where [Formula: see text] is a discrete valuation ring with a finite residue field. We use this to provide a minimal set of generators for the ideal of polynomials in [Formula: see text] that send [Formula: see text] to zero, where [Formula: see text] is a finite commutative local principal ideal ring.

scholarly journals RINGS WHOSE ANNIHILATING-IDEAL GRAPHS HAVE POSITIVE GENUS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250049 ◽  
Author(s):  
F. ALINIAEIFARD ◽  
M. BEHBOODI
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Artinian Ring ◽  
Commutative Rings ◽  
Gorenstein Ring ◽  
Isomorphism Classes ◽  
Artinian Rings ◽  
Vertex Set ◽  
Positive Genus

Let R be a commutative ring and 𝔸(R) be the set of ideals with nonzero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus γ(𝔸𝔾(R)). It is shown that if R is an Artinian ring such that γ(𝔸𝔾(R)) < ∞, then either R has only finitely many ideals or (R, 𝔪) is a Gorenstein ring with maximal ideal 𝔪 and v.dimR/𝔪𝔪/𝔪2= 2. Also, for any two integers g ≥ 0 and q > 0, there are only finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) γ(𝔸𝔾(R)) = g and (ii) |R/𝔪| ≤ q for every maximal ideal 𝔪 of R. Also, it is shown that if R is a non-domain Noetherian local ring such that γ(𝔸𝔾(R)) < ∞, then either R is a Gorenstein ring or R is an Artinian ring with only finitely many ideals.

scholarly journals A Canonical Set For Matrices Over a Principal Ideal Ring Modulo m

1955 ◽  
Vol 7 ◽  
pp. 54-59 ◽  
Author(s):  
L. E. Fuller
Keyword(s):  
Commutative Ring ◽  
Principal Ideal ◽  
Unit Element ◽  
Principal Ideal Ring ◽  
Ideal Ring

If m ∈ P where P is a p.i.r. (principal ideal ring), then P/ {m} is a commutative ring with unit element. The elements of this ring are designated by ā where a ∈ P. The set of square matrices of order n with elements in P/ {m} forms a ring with unit element. The units in this ring are the unimodular matrices, i.e., the matrices whose determinants are units of P/ {m}.

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

2013 ◽  
Vol 12 (04) ◽  
pp. 1250199 ◽  
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).

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

Sign in / Sign up

Export Citation Format

Share Document