Comaximal graph of commutative rings

By H. Maimani, M. Salimi, Asiyeh Sattari, Siamak Yassemi. 2007

159 citations recorded by the source corpus.

Abstract Let R be a commutative ring with identity. Let Γ ( R ) be a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if R a + R b = R . In this paper we consider a subgraph Γ 2 ( R ) of Γ ( R ) which consists of non-unit elements. We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph Γ 2 ( R ) ∖ J ( R ) . In addition, it is shown that for two finite semi-local rings R and S, if R is reduced, then Γ ( R ) ≅ Γ ( S ) if and only if R ≅ S .

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b3ab55de25ed7d8f4374874ec8a286aae576b224

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Graphs Attached to Rings Revisited (2011, unknown): “1 ([24], Theorem 2.”
On the Total Graph and Its Complement of a Commutative Ring (2013, background): “In [14], Sharma and Bhatwadekar defined a graph of a commutative ring R with vertex set as R and two distinct vertices x and y are adjacent if and only if Rx + Ry = R and later it was named as ‘comaximal graph’ (see [12]).”
The Annihilating-Ideal Graph of Commutative Rings I (2008, background): “In [18], Sharma and Bhatwadekar define another graph on R, G(R), with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra+Rb = R (see also [15], in which, the notion of “comaximal graph of commutative rings” is investigated).”
On the structure of comaximal graphs of commutative rings with identity (2011, background): “We are able to give the complete description of comaximal graphs in this case (see [3]).”
Unit Graphs Associated with Rings (2010, background): “This graph is a subgraph of the “comaximal” graph studied in [12].”