Some results about ϕ-primal subsemimodules

2020 ◽  
pp. 2150093
Author(s):  
M. Ebrahimpour
Keyword(s):  
Commutative Ring ◽  
Commutative Semiring ◽  
Nonzero Identity

Let [Formula: see text] be a commutative semiring with nonzero identity and [Formula: see text] an [Formula: see text]-semimodule. In this paper, we introduce the concept of [Formula: see text]-primal subsemimodule of [Formula: see text] that is a generalization of primal ideal of a commutative ring. Then we give some examples and properties of these subsemimodules. Also, some characterizations of [Formula: see text]-primal subsemimodules are presented.

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
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.

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250179 ◽  
Author(s):  
A. AZIMI ◽  
A. ERFANIAN ◽  
M. FARROKHI D. G.
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Independence Number ◽  
Commutative Rings ◽  
Dominating Number ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.

The proof of a conjecture in Jacobson graph of a commutative ring

2015 ◽  
Vol 14 (10) ◽  
pp. 1550107 ◽  
Author(s):  
S. Akbari ◽  
S. Khojasteh ◽  
A. Yousefzadehfard
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Clique Number ◽  
Finite Ring ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. The Jacobson graph of R denoted by 𝔍R is a graph with the vertex set R\J(R), and two distinct vertices x, y ∈ V(𝔍R) are adjacent if and only if 1 - xy ∉ U(R), where U(R) is the set of all unit elements of R. Let ω(𝔍R) denote the clique number of 𝔍R. It was conjectured that if [Formula: see text] is a commutative finite ring and (Ri, 𝔪i) is a local ring, for i = 1, …, n, then [Formula: see text], where Fi = Ri/𝔪i, for i = 1, …, n. In this paper, we prove that if R is a commutative ring (not necessarily finite) and R is not a field, then ω(𝔍R) = max 𝔪∈ Max (R) |𝔪| and using this we show that the aforementioned conjecture holds.

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
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.

scholarly journals On Almost Semiprime Submodules

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Farkhonde Farzalipour
Keyword(s):  
Commutative Ring ◽  
Finitely Generated ◽  
Basic Properties ◽  
Nonzero Identity

We introduce the concept of almost semiprime submodules of unitary modules over a commutative ring with nonzero identity. We investigate some basic properties of almost semiprime and weakly semiprime submodules and give some characterizations of them, especially for (finitely generated faithful) multiplication modules.

When is the Jacobson graph projective?

2018 ◽  
Vol 17 (09) ◽  
pp. 1850168
Author(s):  
Atossa Parsapour ◽  
Khadijeh Ahmadjavaheri
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the Jacobson radical of [Formula: see text]. The Jacobson graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertex-set [Formula: see text], such that two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] is not a unit of [Formula: see text]. The goal in this paper is to list every finite commutative ring [Formula: see text] with nonzero identity (up to isomorphism) such that the graph [Formula: see text] is projective.

ON THE TOTAL GRAPH OF A COMMUTATIVE RING WITHOUT THE ZERO ELEMENT

2012 ◽  
Vol 11 (04) ◽  
pp. 1250074 ◽  
Author(s):  
DAVID F. ANDERSON ◽  
AYMAN BADAWI
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Zero Element ◽  
Induced Subgraphs ◽  
Total Graph ◽  
Zero Divisors ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero-divisors. The total graph of R is the (undirected) graph T(Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we study the two (induced) subgraphs Z0(Γ(R)) and T0(Γ(R)) of T(Γ(R)), with vertices Z(R)\{0} and R\{0}, respectively. We determine when Z0(Γ(R)) and T0(Γ(R)) are connected and compute their diameter and girth. We also investigate zero-divisor paths and regular paths in T0(Γ(R)).

PLANAR, OUTERPLANAR, AND RING GRAPH OF THE COZERO-DIVISOR GRAPH OF A FINITE COMMUTATIVE RING

2012 ◽  
Vol 11 (06) ◽  
pp. 1250103 ◽  
Author(s):  
MOJGAN AFKHAMI ◽  
KAZEM KHASHYARMANESH
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Ring Graph ◽  
Nonzero Identity ◽  
Finite Commutative Rings

Let R be a commutative ring with nonzero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex-set W*(R), which is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b in W*(R) are adjacent if and only if a ∉ Rb and b ∉ Ra. In this paper, we characterize all finite commutative rings R such that Γ′(R) is planar, outerplanar or ring graph.

scholarly journals On 1-absorbing primary ideals of commutative rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050111 ◽  
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]

Characterization of rings with nullity degree at least 1 4

2019 ◽  
Vol 18 (04) ◽  
pp. 1950076
Author(s):  
M. A. Esmkhani ◽  
S. M. Jafarian Amiri
Keyword(s):  
Prime Number ◽  
Commutative Ring ◽  
Finite Commutative Ring ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity. The nullity degree of [Formula: see text], denoted by [Formula: see text], is the probability that the multiplication of two randomly chosen elements of [Formula: see text] is zero. In this paper, we characterize all rings [Formula: see text] with [Formula: see text]. Also, we study rings [Formula: see text] with [Formula: see text], where [Formula: see text] is a prime number.

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