Year: 2026
Type: preprint
Abstract: In analogy with the notion of Krull dimension, for every commutative ring $R$, we introduce the notion of pure dimension $\pdim(R)$ by considering chains of purely-prime ideals. Then we see that the inequality $\pdim(R)\leqslant\dim(R)$ holds for a very broad class of rings $R$, including pseudo-Noetherian rings (e.g., Noetherian rings), Gelfand rings (e.g., local rings), mp-rings (e.g., integral domains), and rings with finitely many maximal ideals or finitely many minimal primes. In fact, for all of these rings $R$, we always have $\pdim(R)=0$, i.e., every purely-prime ideal of $R$ is purely-maximal. \\ However, we prove a classification theorem which states for each ordinal number $Ω$ the proper pure ideals of the commutative ring $$R_Ω=k[X_α : α<Ω]/\big(X_α(1-X_β) : α<β<Ω\big)$$ are exactly of the form $I_λ=(x_α: α<λ)$ with $λ=0$ or $λ\leqslantΩ$ a limit ordinal (we write $x_α$ for the image of $X_α$ in $R_Ω$). In addition, all of them are purely-prime ideals and form the following unique strictly ascending chain: $$I_{0}=0\subset I_ω\subset I_{ω\cdot2}\subset I_{ω\cdot 3}\subset\ldots\subset I_{ω\cdotα}$$ where $α$ is a unique ordinal with $Ω=ω\cdotα+n$ and $n\inω$. To prove this theorem, we use the full strength of ...
Source: arXiv (Cornell University)
Author: Tarizadeh, Abolfazl
Institution: University of Maragheh
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203654670