The exact zero-divisor graph of a reduced ring

2021 ◽  
Author(s):  
S. Visweswaran ◽  
Premkumar T. Lalchandani
Keyword(s):  
Zero Divisor ◽  
Zero Divisor Graph ◽  
Reduced Ring

RECOVERING RINGS FROM ZERO-DIVISOR GRAPHS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350047 ◽  
Author(s):  
SHANE P. REDMOND
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Zero Divisor ◽  
Zero Divisor Graph ◽  
Reduced Ring ◽  
Non Local

Suppose G is the zero-divisor graph of some commutative ring with 1. When G has four or more vertices, a method is presented to find a specific commutative ring R with 1 such that Γ(R) ≅ G. Furthermore, this ring R can be written as R ≅ R1 × R2 × ⋯ × Rn, where each Ri is local and this representation of R is unique up to factors Ri with isomorphic zero-divisor graphs. It is also shown that for graphs on four or more vertices, no local ring has the same zero-divisor graph as a non-local ring and no reduced ring has the same zero-divisor graph as a non-reduced ring.

scholarly journals Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring

2021 ◽  
Vol 29 (2) ◽  
pp. 51-70
Author(s):  
Mehdi Badie
Keyword(s):  
Zero Divisor ◽  
Zariski Topology ◽  
Topological Properties ◽  
Zero Divisor Graph ◽  
Graph Properties ◽  
Reduced Ring ◽  
Isolated Point

Abstract We translate some graph properties of 𝔸𝔾(R) and Γ(R) to some topological properties of Zariski topology. We prove that the facts “(1) The zero ideal of R is an anti fixed-place ideal. (2) Min(R) does not have any isolated point. (3) Rad(𝔸𝔾 (R)) = 3. (4) Rad(Γ(R)) = 3. (5) Γ(R) is triangulated (6) 𝔸𝔾 (R) is triangulated.” are equivalent. Also, we show that if the zero ideal of a ring R is a fixed-place ideal, then dt t (𝔸𝔾 (R)) = |ℬ(R)| and also if in addition |Min(R)| > 2, then dt(𝔸𝔾 (R)) = |ℬ (R)|. Finally, it is shown that dt(𝔸𝔾 (R)) is finite if and only if dt t (𝔸𝔾 (R)) is finite if and only if Min(R) is finite.

scholarly journals Sub-semigroups determined by the zero-divisor graph

2008 ◽  
Vol 308 (22) ◽  
pp. 5122-5135 ◽  
Author(s):  
Tongsuo Wu ◽  
Dancheng Lu
Keyword(s):  
Zero Divisor ◽  
Zero Divisor Graph

ON RINGS R WHOSE GRAPHS Γ(R) SATISFY CONDITION (Kp)

2011 ◽  
Vol 10 (04) ◽  
pp. 665-674
Author(s):  
LI CHEN ◽  
TONGSUO WU
Keyword(s):  
Prime Number ◽  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Satisfy Condition ◽  
Induced Subgraph ◽  
Zero Divisor Graph ◽  
Ring Graph ◽  
Finite Commutative Rings

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).

scholarly journals Zero Divisor Graph for the Ring of Eisenstein Integers Modulo n

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Osama Alkam ◽  
Emad Abu Osba
Keyword(s):  
Zero Divisor ◽  
Zero Divisor Graph

Let En be the ring of Eisenstein integers modulo n. In this paper we study the zero divisor graph Γ(En). We find the diameters and girths for such zero divisor graphs and characterize n for which the graph Γ(En) is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.

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