A Note on Comaximal Graph of Non-commutative Rings |
| |
Authors: | Saieed Akbari Mohammad Habibi Ali Majidinya Raoofe Manaviyat |
| |
Institution: | 1. Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran 2. Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
|
| |
Abstract: | Let R be a ring with unity. The graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra?+?Rb?=?R. Let Γ2(R) be the subgraph of Γ(R) induced by the non-unit elements of R. Let R be a commutative ring with unity and let J(R) denote the Jacobson radical of R. If R is not a local ring, then it was proved that: - If $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph, then n?=?2.
- If there exists a vertex of $\Gamma_2(R)\backslash J(R)$ which is adjacent to every vertex, then R????2×F, where F is a field.
In this note we generalize the above results to non-commutative rings and characterize all non-local ring R (not necessarily commutative) whose $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|