Pure dimension and counterexamples to the maximality of purely-primes
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)leqslantdim(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 RΩ=k[Xα : α<Ω]/(Xα(1−Xβ) : α<β<Ω) are exactly of the form Iλ=(xα: α<λ) with λ=0 or λleqslantΩ a limit ordinal (we write xα for the image of Xα in RΩ). In addition, all of them are purely-prime ideals and form the following unique strictly ascending chain: I0=0⊂ Iω⊂ Iω·2⊂ Iω· 3⊂…⊂ Iω·α where α is a unique ordinal with Ω=ω·α+n and n∈ω. To prove this theorem, we use the full strength of ...
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BibTeX
@misc{tarizadeh2026pure,
title = {{Pure dimension and counterexamples to the maximality of purely-primes}},
author = {Abolfazl Tarizadeh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.14251},
eprint = {2608.14251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14251},
}
RIS
TY - UNPB
TI - Pure dimension and counterexamples to the maximality of purely-primes
AU - Abolfazl Tarizadeh
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2608.14251
UR - https://arxiv.org/abs/2608.14251
AB - 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)leqslantdim(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]/(X_α(1−X_β) : α<β<Ω) are exactly of the form I_λ=(x_α: α<λ) with λ=0 or λleqslantΩ 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⊂ I_ω⊂ I_ω·2⊂ I_ω· 3⊂…⊂ I_ω·α where α is a unique ordinal with Ω=ω·α+n and n∈ω. To prove this theorem, we use the full strength of ...
ER -