THE MATRIX JACOBSON GRAPH OF LOCAL COMMUTATIVE RINGS
The notion of Jacobson graph and n-array Jacobson graph of a commutative ring were introduced in 2012 and 2018, respectively, by Azimi et al. In this paper we generalize them into matrix Jacobson graph of ring. Let R be a commutative ring, U(R) be the unit grup of R, J(R) be the Jacobson radical...
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| Format: | Theses |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/46507 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | The notion of Jacobson graph and n-array Jacobson graph of a commutative ring
were introduced in 2012 and 2018, respectively, by Azimi et al. In this paper we
generalize them into matrix Jacobson graph of ring. Let R be a commutative ring,
U(R) be the unit grup of R, J(R) be the Jacobson radical of R and R^(m x n) be the matrix
of size m x n over R. The matrix Jacobson graph of size m x n over ring R, denoted
J_R^(m x n), is defined as a graph with vertex set R^(m x n)\ (J(R))^(m x n) such that two distinct
vertices A, B are adjacent if and only if 1-det(A^t B) not in U(R). We study this class
of graphs for local ring, determine the number of components and the diameter of
connected component of the square and non square matrix Jacobson graph of local
ring. |
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