Hamiltonian trace graph of matrices

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a positive integer and [Formula: see text] be the set of all [Formula: see text] matrices over [Formula: see text] For a matrix [Formula: see text] Tr[Formula: see text] is the trace of [Formula: see text] The trace graph of the matrix ring [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text][Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] The ideal-based trace graph of the matrix ring [Formula: see text] with respect to an ideal [Formula: see text] of [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] In this paper, we investigate some properties and structure of [Formula: see text] Further, it is proved that both [Formula: see text] and [Formula: see text] are Hamiltonian.

On the relation between torsion submodule and Fitting ideals

10.1142/s0219498822501730 ◽

2021 ◽

pp. 2250173

Author(s):

S. Hadjirezaei

Keyword(s):

Commutative Ring ◽

Local Ring ◽

Principal Ideal ◽

Index Set ◽

Constructive Description ◽

Regular Ideal ◽

Fitting Ideal ◽

The Matrix ◽

Matrix Formula ◽

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a submodule of [Formula: see text] which consists of columns of a matrix [Formula: see text] with [Formula: see text] for all [Formula: see text], [Formula: see text], where [Formula: see text] is an index set. For every [Formula: see text], let I[Formula: see text] be the ideal generated by subdeterminants of size [Formula: see text] of the matrix [Formula: see text]. Let [Formula: see text]. In this paper, we obtain a constructive description of [Formula: see text] and we show that when [Formula: see text] is a local ring, [Formula: see text] is free of rank [Formula: see text] if and only if I[Formula: see text] is a principal regular ideal, for some [Formula: see text]. This improves a lemma of Lipman which asserts that, if [Formula: see text] is the [Formula: see text]th Fitting ideal of [Formula: see text] then [Formula: see text] is a regular principal ideal if and only if [Formula: see text] is finitely generated free and [Formula: see text] is free of rank [Formula: see text]

Some results on a spanning subgraph of the complement of the annihilating-ideal graph of a commutative reduced ring

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500125 ◽

2019 ◽

Vol 11 (01) ◽

pp. 1950012

Author(s):

S. Visweswaran ◽

Anirudhdha Parmar

Keyword(s):

Commutative Ring ◽

Integral Domain ◽

Undirected Graph ◽

Ideal Formula ◽

Graph Theoretic ◽

Spanning Subgraph ◽

Reduced Ring ◽

Vertex Set

Let [Formula: see text] be a commutative ring with identity which is not an integral domain. Let [Formula: see text] denote the set of all annihilating ideals of [Formula: see text] and let us denote [Formula: see text] by [Formula: see text]. For an ideal [Formula: see text] of [Formula: see text], we denote the annihilator of [Formula: see text] in [Formula: see text] by [Formula: see text]. That is, [Formula: see text]. In this note, for any ring [Formula: see text] with [Formula: see text], we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] are joined by an edge if and only if either [Formula: see text] or [Formula: see text]. Let [Formula: see text] be a reduced ring. The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-theoretic properties of [Formula: see text].

Associate Ring Graphs

Mapana Journal of Sciences ◽

10.12723/mjs.16.5 ◽

2010 ◽

Vol 9 (1) ◽

pp. 31-40

Author(s):

M. James Subhakar

Keyword(s):

Positive Integer ◽

Commutative Ring ◽

Equivalence Relation ◽

Undirected Graph ◽

Ring Of Integers ◽

Vertex Set ◽

Ring Graph

R is a commutative ring with unity. The associate ring graph AG(R) is the graph with the vertex set V = R • {0} and edge set E = {(a, b) / a, b are associates and a ≠ b}. Since the relation of being associate is on equivalence relation, this graph is an undirected graph and also each component is complete. In this paper, I present some of the interesting results majority of which are for the ring of integers modulo n, n is a positive integer.

Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings

ISRN Discrete Mathematics ◽

10.5402/2011/459547 ◽

2011 ◽

Vol 2011 ◽

pp. 1-7

Author(s):

Shahabaddin Ebrahimi Atani ◽

Ahmad Yousefian Darani

Keyword(s):

Commutative Ring ◽

Zero Divisor ◽

Undirected Graph ◽

Noncommutative Rings ◽

Zero Divisor Graph ◽

Vertex Set ◽

Definition Of ◽

Let R be a commutative ring and I an ideal of R. The zero-divisor graph of R with respect to I, denoted ΓI(R), is the undirected graph whose vertex set is {x∈R∖I|xy∈I for some y∈R∖I} with two distinct vertices x and y joined by an edge when xy∈I. In this paper, we extend the definition of the ideal-based zero-divisor graph to noncommutative rings.

A generalized ideal-based zero-divisor graph

10.1142/s0219498815500796 ◽

2015 ◽

Vol 14 (06) ◽

pp. 1550079 ◽

Cited By ~ 1

Author(s):

M. J. Nikmehr ◽

S. Khojasteh

Keyword(s):

Prime Ideal ◽

Commutative Ring ◽

Chromatic Number ◽

Zero Divisor ◽

Undirected Graph ◽

Clique Number ◽

Proper Ideal ◽

Graph Structure ◽

Zero Divisor Graph ◽

Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

Complementation of the Jacobson group in a matrix ring

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700035103 ◽

2005 ◽

Vol 72 (2) ◽

pp. 317-324

Author(s):

David Dolžan

Keyword(s):

Normal Subgroup ◽

Commutative Ring ◽

Jacobson Radical ◽

Matrix Ring ◽

Graded Rings ◽

Matrix Rings ◽

Generators And Relations ◽

Group Of Units ◽

The Matrix ◽

Finite Commutative Ring

The Jacobson group of a ring R (denoted by  = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

Generalized unit and unitary Cayley graphs of finite rings

10.1142/s0219498819500063 ◽

2019 ◽

Vol 18 (01) ◽

pp. 1950006 ◽

Cited By ~ 2

Author(s):

T. Tamizh Chelvam ◽

S. Anukumar Kathirvel

Keyword(s):

Commutative Ring ◽

Undirected Graph ◽

Unit Element ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Finite Rings ◽

Vertex Set ◽

Finite Commutative Ring ◽

Necessary And Sufficient ◽

Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

Derivations of some classes of matrix rings

10.1142/s021949881750027x ◽

2017 ◽

Vol 16 (02) ◽

pp. 1750027 ◽

Cited By ~ 2

Author(s):

Feride Kuzucuoğlu ◽

Umut Sayın

Keyword(s):

Associative Ring ◽

Matrix Ring ◽

Matrix Rings ◽

Ideal Formula ◽

The Matrix

Let [Formula: see text] be the ring of all (lower) niltriangular [Formula: see text] matrices over any associative ring [Formula: see text] with identity and [Formula: see text] be the ring of all [Formula: see text] matrices over an ideal [Formula: see text] of [Formula: see text]. We describe all derivations of the matrix ring [Formula: see text].

Amount algebras

10.1142/s021949882250102x ◽

2021 ◽

pp. 2250102

Author(s):

Peyman Nasehpour

Keyword(s):

Positive Integer ◽

Valuation Ring ◽

Prüfer Domain ◽

Radical Ideal ◽

Ideal Formula ◽

Torsion Free ◽

In this paper, as a generalization to content algebras, we introduce amount algebras. Similar to the Anderson–Badawi [Formula: see text] conjecture, we prove that under some conditions, the formula [Formula: see text] holds for some amount [Formula: see text]-algebras [Formula: see text] and some ideals [Formula: see text] of [Formula: see text], where [Formula: see text] is the smallest positive integer [Formula: see text] that the ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing. A corollary to the mentioned formula is that if, for example, [Formula: see text] is a Prüfer domain or a torsion-free valuation ring and [Formula: see text] is a radical ideal of [Formula: see text], then [Formula: see text].

On 1-absorbing primary ideals of commutative rings

10.1142/s021949882050111x ◽

2019 ◽

Vol 19 (06) ◽

pp. 2050111 ◽

Cited By ~ 2

Author(s):

Ayman Badawi ◽

Ece Yetkin Celikel

Keyword(s):

Positive Integer ◽

Prime Ideal ◽

Commutative Ring ◽

Commutative Rings ◽

Proper Ideal ◽

Primary Ideal ◽

Ideal Formula ◽

Noetherian Domain ◽

Primary Ideals ◽

Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text]