The ring $R\{X\}$

Year 2024, Early Access, 1 - 8

Emad Abuosba Mariam Al-azaizeh

https://doi.org/10.24330/ieja.1478925

Abstract

Let $R$ be a commutative ring with unity and $W=\{f(X)\in R[X]:f(0)=1\}$. We define $R\{X\}=W^{-1}R[X]$. We show that the maximal ideals of $R\{X\} $ are of the form $W^{-1}(M,X)$ where $M$ is a maximal ideal of $R$, and so if $R$ is finite dimensional, then $\dim R\{X\}=\dim R[X]$. We show that $R\{X\}$ is a Prüfer ring if and only if $R$ is a von Neumann regular ring, and so if $R\{X\}$ satisfies one of the Prüfer conditions, it satisfies all of them.

Keywords

Prime ideal, localization, over ring, polynomial ring, Prüfer conditions

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