Prime ideals in infinite products of commutative rings

Carmelo A. Finocchiaro; Sophie Frisch; Daniel Windisch

doi:10.1142/S0219199723500451

Prime ideals in infinite products of commutative rings

Carmelo A. Finocchiaro^*, Sophie Frisch, Daniel Windisch

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

We describe the prime ideals and, in particular, the maximal ideals in products R = Dλ of families (Dλ)λ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra (max(Dλ)), where max(Dλ) is the spectrum of maximal ideals of Dλ, and denotes the power set. If every Dλ is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of R. If every Dλ is a Prüfer domain, we completely characterize all prime ideals of R.

Original language	English
Article number	2350045
Journal	Communications in Contemporary Mathematics
Early online date	2023
DOIs	https://doi.org/10.1142/S0219199723500451
Publication status	E-pub ahead of print - 2023

Keywords

prime ideals
Product rings
Prüfer domains
ultrafilters

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1142/S0219199723500451

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abstract = "We describe the prime ideals and, in particular, the maximal ideals in products R = Dλ of families (Dλ)λ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra (max(Dλ)), where max(Dλ) is the spectrum of maximal ideals of Dλ, and denotes the power set. If every Dλ is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of R. If every Dλ is a Pr{\"u}fer domain, we completely characterize all prime ideals of R.",

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author = "Finocchiaro, {Carmelo A.} and Sophie Frisch and Daniel Windisch",

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T1 - Prime ideals in infinite products of commutative rings

AU - Finocchiaro, Carmelo A.

AU - Frisch, Sophie

AU - Windisch, Daniel

PY - 2023

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N2 - We describe the prime ideals and, in particular, the maximal ideals in products R = Dλ of families (Dλ)λ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra (max(Dλ)), where max(Dλ) is the spectrum of maximal ideals of Dλ, and denotes the power set. If every Dλ is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of R. If every Dλ is a Prüfer domain, we completely characterize all prime ideals of R.

AB - We describe the prime ideals and, in particular, the maximal ideals in products R = Dλ of families (Dλ)λ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra (max(Dλ)), where max(Dλ) is the spectrum of maximal ideals of Dλ, and denotes the power set. If every Dλ is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of R. If every Dλ is a Prüfer domain, we completely characterize all prime ideals of R.

KW - prime ideals

KW - Product rings

KW - Prüfer domains

KW - ultrafilters

UR - http://www.scopus.com/inward/record.url?scp=85179080313&partnerID=8YFLogxK

U2 - 10.1142/S0219199723500451

DO - 10.1142/S0219199723500451

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SN - 0219-1997

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

M1 - 2350045

ER -

Prime ideals in infinite products of commutative rings

Abstract

Keywords

ASJC Scopus subject areas

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