Prime ideals in infinite products of commutative rings

2020/09/07 by Finocchiaro, Carmelo A., Frisch, Sophie, Windisch, Daniel
#13A15 #13C13 #13F05 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2009.03069

Abstract

We describe the prime ideals and, in particular, the maximal ideals in products R = ∏ D_λ of families (D_λ)_{λ∈ Λ} of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra ∏ P(max(D_λ)), where max(D_λ) is the spectrum of maximal ideals of D_λ, and P denotes the power set. If every D_λ 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 D_λ is a Prüfer domain, we completely characterize all prime ideals of R.

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