GENERALIZATION OF JACOBSON GRAPH OF RINGS - Humaira, Siti
Jacobson graph and Jacobson n-array graph of the commutative ring were first introduced in 2013 and 2018 by Azimi et al. In this dissertation, we discuss the generalization of the Jacobson graph. The first generalization is called the Jacobson matrix graph. Let R be a commutative ring, U(R) be an unit group of R, J(R) be a Jacobson radical of R and Rm×n be a matrix set of size m × n over R. The Jacobson matrix graph of size m× n over rings R, denoted J(R)m×n, is defined as a graph with set of points Rm×n \ (J(R))m×n such that two distinct points A,B are adjacent if and only if 1 ? (AtB) /? U(R). The Second generalization is Jacobson graphs of non-commutative rings with specific cases on ring matrices. The purpose of this study is to enrich the properties of the Jacobson graph over the rings by discussing the non-commutative aspects of the ring and completing the generalization properties of the matrix Jacobson graph. This dissertation has succeeded in generalizing the properties of the Jacobson graph. The matrix Jacobson graph’s properties of connectivity, diameter, planarity, and perfectness have been obtained on the matrix Jacobson graph over fields, local rings, and nonlocal rings. In the Jacobson graph of the ring matrix, we obtained results like diameter, the relationship between the degree of a vertex and the dimensions of the column, and some special properties of the triangular matrix. The research method are used by exploring and adapting the results obtained previously, including utilizing the properties and structure of linear operators in finite-dimensional vector spaces and the relation between modules and quotient modules of their submodules.