Abstract
Let \(R\) be a commutative ring with a non-zero identity and \(\mathfrak {J}_R\) be its Jacobson graph. We show that if \(R\) and \(R'\) are finite commutative rings, then \(\mathfrak {J}_R\cong \mathfrak {J}_{R'}\) if and only if \(|J(R)|=|J(R')|\) and \(R/J(R)\cong R'/J(R')\). Also, for a Jacobson graph \(\mathfrak {J}_R\), we obtain the structure of group \(\mathrm {Aut}(\mathfrak {J}_R)\) of all automorphisms of \(\mathfrak {J}_R\) and prove that under some conditions two semi-simple rings \(R\) and \(R'\) are isomorphic if and only if \(\mathrm {Aut}(\mathfrak {J}_R)\cong \mathrm {Aut}(\mathfrak {J}_{R'})\).
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Acknowledgments
The authors would like to thank Prof. Peter M. Neumann for pointing out that the groups in Lemma 3.6 have different orders except for few possibilities.
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Azimi, A., Erfanian, A. & Farrokhi D. G., M. Isomorphisms between Jacobson graphs. Rend. Circ. Mat. Palermo 63, 277–286 (2014). https://doi.org/10.1007/s12215-014-0157-7
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DOI: https://doi.org/10.1007/s12215-014-0157-7