The M-Regular Graph of a Commutative Ring
-
M. J. Nikmehr
and
F. Heydari
Abstract
Let R be a commutative ring and M be an R-module, and let Z(M) be the set of all zero-divisors on M. In 2008, D. F. Anderson and A. Badawi introduced the regular graph of R. In this paper, we generalize the regular graph of R to the M-regular graph of R, denoted by M-Reg(Γ(R)). It is the undirected graph with all M-regular elements of R as vertices, and two distinct vertices x and y are adjacent if and only if x+y ∈ Z(M). The basic properties and possible structures of M-Reg(Γ(R)) are studied. We determine the girth of the M-regular graph of R. Also, we provide some lower bounds for the independence number and the clique number of M- Reg(Γ(R)). Among other results, we prove that for every Noetherian ring R and every finitely generated module M over R, if 2 ∉ Z(M) and the independence number of M-Reg(Γ(R)) is finite, then R is finite.
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© 2015 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- The M-Regular Graph of a Commutative Ring
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- Sharp Inequalities for Polygamma Functions
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- Approximate Higher Ring Derivations in Non-Archimedean Banach Algebras
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Articles in the same Issue
- The M-Regular Graph of a Commutative Ring
- Further Properties of the Lattice of Torsion Classes of Abelian Cyclically Ordered Groups
- Free Power-Commutative Groupoids
- On Systems of Independent Sets
- Note on G-Module Structure of Orders
- Semisimple Hopf Algebras of Dimension 2q3
- On the Convergence of a Mapped by a Function
- Multiple Solutions of Nonlinear Fractional Differential Equations with p-Laplacian Operator and Nonlinear Boundary Conditions
- Meromorphic Functions Sharing Pairs of Small Functions
- Sharp Inequalities for Polygamma Functions
- Existence Results for Some Higher-Order Evolution Equations with Time-Dependent Unbounded Operator Coefficients
- Existence of Ground State Solutions for Hamiltonian Elliptic Systems with Gradient Terms
- Approximate Higher Ring Derivations in Non-Archimedean Banach Algebras
- Characterizations of Banach Spaces which are not Isomorphic to any of their Proper Subspaces
- Operator Inequalities Related to Q-Class Functions
- On a Class of Locally Dually Flat (α,β)–Metrics
- Limit Theorems for the Counting Function of Eigenvalues up to Edge in Covariance Matrices
- A Generalization of Jakubec's Formula
