On the diameter of compressed zero-divisor graphs of Ore extensions
Abstract
This paper continues the ongoing effort to study the compressed zero-divisor graph over non-commutative rings.
The purpose of our paper is to study
the diameter of the compressed zero-divisor graph of Ore extensions and give a complete characterization of the possible diameters of $\Gamma_{E} \big(R[x; \alpha, \delta]\big)$, where the base ring $R$ is reversible and also have the $(\alpha , \delta)$-compatible property. Also, we give a complete characterization of the diameter of $\Gamma_{E}\big( R[[x; \alpha]]\big) $, where $R$ is a reversible, $\alpha$-compatible and right Noetherian ring. By some examples, we show that all of the assumptions ``reversiblity'', ``$(\alpha,\delta)$-compatiblity" and ``Noetherian" in our main results are crucial.
The purpose of our paper is to study
the diameter of the compressed zero-divisor graph of Ore extensions and give a complete characterization of the possible diameters of $\Gamma_{E} \big(R[x; \alpha, \delta]\big)$, where the base ring $R$ is reversible and also have the $(\alpha , \delta)$-compatible property. Also, we give a complete characterization of the diameter of $\Gamma_{E}\big( R[[x; \alpha]]\big) $, where $R$ is a reversible, $\alpha$-compatible and right Noetherian ring. By some examples, we show that all of the assumptions ``reversiblity'', ``$(\alpha,\delta)$-compatiblity" and ``Noetherian" in our main results are crucial.
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