Moreover, we consider some topological conditions for NJ rings and show for a NJ ring $R$ that $R$ is J-pm if and only if $J$-$Spec(R)$ is a normal space if and only if $Max(R)$ is a retract of $J$-$Spec(R)$.
📌 3 Key Takeaways
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Objective & Hypothesis:A ring $R$ is said to be NJ if $J(R)=N(R)$.
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Key Findings & Data:At the same time, we also prove that if $R$ is a weakly 2-primal $\alpha$-compatible ring with an isomorphism $\alpha$ of $R$, then $R[x;\alpha]$ is NJ; if $R$ is a weakly 2-primal $\delta$-compatible ring with a derivation $\delta$ of $R$, then $R[x;\delta]$ is NJ.
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Conclusion & Impact:Moreover, we consider some topological conditions for NJ rings and show for a NJ ring $R$ that $R$ is J-pm if and only if $J$-$Spec(R)$ is a normal space if and only if $Max(R)$ is a retract of $J$-$Spec(R)$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Jacobson radical · nilpotent element · weakly 2-primal ring
Abstract
A ring $R$ is said to be NJ if $J(R)=N(R)$. This paper mainly studies the relationship between NJ rings and related rings, and investigates the Dorroh extension, the Nagata extension, the Jordan extension, and some other extensions of NJ rings. At the same time, we also prove that if $R$ is a weakly 2-primal $\alpha$-compatible ring with an isomorphism $\alpha$ of $R$, then $R[x;\alpha]$ is NJ; if $R$ is a weakly 2-primal $\delta$-compatible ring with a derivation $\delta$ of $R$, then $R[x;\delta]$ is NJ. Moreover, we consider some topological conditions for NJ rings and show for a NJ ring $R$ that $R$ is J-pm if and only if $J$-$Spec(R)$ is a normal space if and only if $Max(R)$ is a retract of $J$-$Spec(R)$.
JIANG, M., WANG, Y., REN, Y. (2019). Extensions and topological conditions of NJ rings. Turkish Journal of Mathematics, Cilt: 43 - Sayı: 1, 44–62. https://ajindex.com/articles/212744