Mixed polynomial/power series rings and relatinos among their spectra

Heinzer, William; Rotthaus, Christel; Wiegand, Sylvia

doi:10.1007/978-0-387-36717-0_14

Cited by 11 publications

(10 citation statements)

References 7 publications

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“…We were motivated to consider generic fiber rings for the embeddings displayed above because of questions related to [4,5] and ultimately because of the following question posed by Melvin Hochster.…”

Section: Introductionmentioning

confidence: 99%

Generic fiber rings of mixed power series/polynomial rings

2006

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Let K be a field, m and n positive integers, and X = {x 1 , . . . , x n }, and Y = {y 1 , . . . , y m } sets of independent variables over K. Let A be the localized polynomial ring K[X] (X) . We prove that every prime ideal P in A = KJXK that is maximal with respect to P ∩ A = (0) has height n − 1. We consider the mixed power series/polynomial rings B := KJXK[Y ] (X,Y ) and C := K[Y ] (Y ) JXK. For each prime ideal P of B = C that is maximal with respect to either P ∩ B = (0) or P ∩ C = (0), we prove that P has height n + m − 2. We also prove each prime ideal P of KJX, Y K that is maximal with respect to P ∩ KJXK = (0) is of height either m or n + m − 2.

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Section: Introductionmentioning

confidence: 99%

Generic fiber rings of mixed power series/polynomial rings

2006

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“…Next, we examine the effect of "partial completions" (completions at a non-maximal ideal) on spectra of rings. Previous work in [6] shows that such rings need not have uniform cardinalities. That is, different varieties of the prime spectrum can have different cardinalities; natural examples arise in the case of mixed polynomial-power series rings.…”

Section: Introductionmentioning

confidence: 99%

Cardinalities of Prime Spectra of Precompletions

Barrett¹,

Graf²,

Loepp³

et al. 2019

Preprint

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Given a complete local (Noetherian) ring T , we find necessary and sufficient conditions on T such that there exists a local domain A with |A| < |T | and A = T , where A denotes the completion of A with respect to its maximal ideal. We then find necessary and sufficient conditions on T such that there exists a domain A with A = T and |Spec(A)| < |Spec(T )|. Finally, we use "partial completions" to create local rings A with A = T such that Spec(A) has varying cardinality in different varieties.

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“…In the diagram α is the cardinality of the set of maximal ideals of R. We first use a remark from [8]. …”

Section: Algebra Then Spec(r) Satisfies (P5) and Hence Is Order-isommentioning

confidence: 99%

Prime Ideals in Noetherian Rings: A Survey

Wiegand¹,

Wiegand²

2010

Ring and Module Theory

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Abstract. We consider the structure of the partially ordered set of prime ideals in a Noetherian ring. The main focus is Noetherian two-dimensional integral domains that are rings of polynomials or power series. IntroductionThe authors have been captivated by the partially ordered set of prime ideals for about four decades. Their initial motivation was interest in Kaplansky's question, Key words and phrases. prime ideals, Noetherian. The authors thank the University of Nebraska for its support of our research projects.

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Mixed polynomial/power series rings and relatinos among their spectra

Cited by 11 publications

References 7 publications

Generic fiber rings of mixed power series/polynomial rings

Generic fiber rings of mixed power series/polynomial rings

Cardinalities of Prime Spectra of Precompletions

Prime Ideals in Noetherian Rings: A Survey

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