The ring $R\{X\}$

Year 2025, Volume: 37 Issue: 37, 36 - 43, 14.01.2025

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

References

• D. D. Anderson, D. F. Anderson and R. Markanda, The rings $R(X)$ and $R\left\langle X\right\rangle$, J. Algebra, 95(1) (1985), 96-115.
• R. Gilmer, Multiplicative Ideal Theory, Pure and Applied Mathematics, 12, Marcel Dekker, Inc., New York, 1972.
• J. A. Huckaba, Commutative Rings with Zero Divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
• J. A. Huckaba and I. J. Papick, Quotient rings of polynomial rings, Manuscripta Math., 31 (1980), 167-196.
• J. A. Huckaba and I. J. Papick, A localization of $R[x]$, Canadian J. Math., 33(1) (1981), 103-115.
• M. Jarrar and S. Kabbaj, Prüfer conditions in the Nagata ring and the Serre's conjecture ring, Comm. Algebra, 46(5) (2018), 2073-2082.
• I. Kaplansky, Commutative Rings, Revised Edition, University of Chicago Press, Chicago, 1974.
• L. R. le Riche, The ring $R\left\langle X\right\rangle$, J. Algebra, 67 (1980), 327-341.