Some results on the intersection graph of submodules of a module
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Saieed Akbari
Abstract
Let R be a ring with identity and M be a unitary left R-module. The intersection graph of submodules of M, denoted by G(M), is defined to be a graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have non-zero intersection. In this paper, we consider the intersection graph of submodules of a module. We determine the structure of modules whose clique numbers are finite. We show that if 1 < ω(G(M)) < ∞, then M is a direct sum of a finite module and a cyclic module, where ω(G(M)) denotes the clique number of G(M). We prove that if ω(G(M)) is not finite, then M contains an infinite clique. Among other results, it is shown that a Noetherian R-module whose intersection of all non-trivial submodules is non-zero, is Artinian.
The authors would like to thank the referee for his/her careful review and valuable comments which helped to improve the paper. The first author is indebted to the School of Mathematics, Institute for Research in Fundamental Sciences (IPM) for support. The research of the first author was in part supported by a grant from IPM (No. 93050212).
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© 2017 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- 10.1515/ms-2015-0200
- Zero-divisor graphs of lower dismantlable lattices I
- Some results on the intersection graph of submodules of a module
- Class number parities of compositum of quadratic function fields
- Examples of beurling prime systems
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Connection between multiplication theorem for Bernoulli polynomials and first factor
- On permutational invariance of the metric discrepancy results
- Evaluation of sums containing triple aerated generalized Fibonomial coefficients
- Linear algebraic proof of Wigner theorem and its consequences
- A note on groups with finite conjugacy classes of subnormal subgroups
- Groups with the same complex group algebras as some extensions of psl(2, pn)
- Klee-Phelps convex groupoids
- On analytic functions with generalized bounded Mocanu variation in conic domain with complex order
- Weak interpolation for the lipschitz class
- Generalized Padé approximants for plane condenser and distribution of points
- Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials
- Positive solutions of nonlocal integral BVPS for the nonlinear coupled system involving high-order fractional differential
- Existence of positive solutions for a nonlinear nth-order m-point p-Laplacian impulsive boundary value problem
- Dirichlet boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses
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- Homoclinic solutions for ordinary (q, p)-Laplacian systems with a coercive potential
- Semi-equivelar maps on the torus and the Klein bottle with few vertices
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