Krull dimension of polynomial rings

Brewer, J. W.; Montgomery, P. R.; Rutter, Edgar; Heinzer, William

doi:10.1007/bfb0068916

Cited by 37 publications

(28 citation statements)

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“…Denote by q = Q ∩ T and p = P ∩ R. We get easily p = q ∩ R and I ⊆ q. Using the fact that R and T are locally Jaffard, it follows from the special chain theorem [6] that: …”

Section: Resultsmentioning

confidence: 99%

Pullbacks and universal catenarity

Jarboui¹,

Jerbi²

2008

Publicacions Matemàtiques

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“…Denote by q = Q ∩ T and p = P ∩ R. We get easily p = q ∩ R and I ⊆ q. Using the fact that R and T are locally Jaffard, it follows from the special chain theorem [6] that: …”

Section: Resultsmentioning

confidence: 99%

Pullbacks and universal catenarity

Jarboui¹,

Jerbi²

2008

Publicacions Matemàtiques

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“…We study the structure of the prime spectrum of T , clarifying the relation between the spectrum of T and those of R and R [X]. We generalize some well-known results previously established for polynomial rings [9]. The final aim of this section is to prove our promised results which state that if R is a locally Jaffard (resp., an S-) domain, then each pseudo-polynomial ring is locally Jaffard (resp., S).…”

Section: In Section 2 We Begin With a Description Of The Valuative Hementioning

confidence: 99%

“…We assume the result for all k < m, where m ≥ 0 and ht(p) = m. To prove that ht( By combining Proposition 2.1 and Lemma 2.10, we have the following theorem which generalizes the special chain theorem ( [9], [19]) and the valuative special chain theorem [15] for a given pseudo-polynomial ring.…”

Section: Proof Of Proposition 27 the Prime Ideal (X) ∩ T Is Nonzeromentioning

confidence: 99%

Absolutely S-domains and pseudo-polynomial rings

Jarboui

Yengui

2002

Colloq. Math.

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Abstract.A domain R is called an absolutely S-domain (for short, AS-domain) if each domain T such that R ⊆ T ⊆ qf(R) is an S-domain. We show that R is an AS-domain if and only if for each valuation overring V of R and each height one prime ideal q of V , the extension R/(q ∩ R) ⊆ V /q is algebraic. A Noetherian domain R is an AS-domain if and only if dim(R) ≤ 1. In Section 2, we study a class of R-subalgebras of R [X] which share many spectral properties with the polynomial ring R[X] and which we call pseudo-polynomial rings. Section 3 is devoted to an affirmative answer to D. E. Dobbs's question of whether a survival pair must be a lying-over pair in the case of transcendental extension. Introduction.In this paper, all rings considered are commutative with identity. An inclusion of rings signifies that the smaller ring is a subring of the larger and has the same identity. Let R be a ring and n a positive integer. We denote by R[n] the ring of polynomials in n indeterminates over R and by R[X] the ring of polynomials in one indeterminate. We denote by dim(R) the Krull dimension of R and by dim v (R) its valuative dimension, that is, the limit of the sequence (dim(R[n]) − n, n ∈ N). If p is a prime ideal of R, we denote by ht(p) the height of p, and by ht v (p) the limit of the sequence (ht(p[n]), n ∈ N).Given a finite-dimensional ring R, we say that R is a Jaffard ring if dim(R) = dim v (R) [2]. This property is not local; we say that R is a locally Jaffard ring if R p is a Jaffard ring for each prime ideal p of R. A domain R is said to be an S-domain if for each height 1 prime ideal p of R, we have ht(p[X]) = 1. A strong S-ring is a ring R such that for each prime ideal p of R, R/p is an S-domain; equivalently for any consecutive primes p ⊂ q in R,. An overring of a domain R is a ring contained between R and its quotient field qf(R).For an extension of domains R ⊆ T , we denote by tr.deg[T : R] the transcendence degree of qf(T ) over qf(R). Recall that an extension R ⊆ T 2000 Mathematics Subject Classification: Primary 13B02; Secondary 13C15, 13A17, 13A18, 13B25, 13E05.

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“…The strong S-property is not stable, in general under polynomial extensions (cf. [6]). In [19], Malik and Mott, defined and studied the stably strong S-domains.…”

Section: Introductionmentioning

confidence: 99%

Universally Catenarian Integral Domains, Strong S-Domains and Semistar Operations

Sahandi

2010

Communications in Algebra

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Abstract. Let D be an integral domain and ⋆ a semistar operation stable and of finite type on it. In this paper, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over D. We introduce and investigate the notions of ⋆-universally catenarian and ⋆-stably strong S-domains and prove that, every ⋆-locally finite dimensional Prüfer ⋆-multiplication domain is ⋆-universally catenarian, and this implies ⋆-stably strong S-domain. We also give new characterizations of ⋆-quasi-Prüfer domains introduced recently by Chang and Fontana, in terms of these notions. IntroductionThe concepts of S(eidenberg)-domains and strong S-domains are crucial ones and were introduced by Kaplansky [18, Page 26]. Recall that an integral domain D is an S-domain if for each prime ideal P of D of height one the extension P D[X] to the polynomial ring in one variable is also of height one. A strong S-domain is a domain D such that, D/P is an S-domain, for each prime P of D. One of the reasons why Kaplansky introduced the notion of strong S-domain was to treat the classes of Noetherian domains and Prüfer domains in a unified frame. Moreover, if D belongs to one of the two classes of domains, then the following dimension formula holds: dim(D[X 1 , · · · , X n ]) = n + dim(D) (cf., [23, Theorem 9] and [24, Theorem 4] is a strong S-domain for each n ≥ 1. Note that the class of Jaffard domains contains the class of stably strong S-domains. The class of stably strong S-domains contains an important class of universally catenarian domains. Recall that a domain D, is called catenarian, if for each pair P ⊂ Q of prime ideals of D, any two saturated chain of prime ideals between P and Q have the same finite length. If for each n ≥ 1, the polynomial ring D[X 1 , · · · , X n ] is catenary, then D is said to be universally catenarian (cf. [4,3]).For several decades, star operations, as described in [17, Section 32], have proven to be an essential tool in multiplicative ideal theory, for studying various classes of 2000 Mathematics Subject Classification. Primary 13G05, 13B24, 13A15, 13C15. Key words and phrases. Semistar operation, star operation, Krull dimension, strong S-domain, Jaffard domain, universally catenarian, catenary. This manuscript is a sequel to [22]. Given a semistar operation ⋆ on D and let ⋆ be the stable semistar operation of finite type canonically associated to ⋆ (the definitions are recalled later in this section), it is possible to define a semistar operation stable and of finite typefor each positive integer n. Every ⋆-Noetherian and P⋆MDs are ⋆-Jaffard domains (cf. [22]). In this paper we define and study two subclass of ⋆-Jaffard domains. Namely in Sections 2 and 3, we define and study ⋆-stably strong S-domains and ⋆-universally catenarian domains. In Section 4 we give new characterizations of ⋆-quasi-Prüfer domains in terms of ⋆-stably strong S-domains and ⋆-universally catenarian domains.To facilitate the reading of the introduction and of the paper, we first review some basic facts on semi...

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Krull dimension of polynomial rings

Cited by 37 publications

References 4 publications

Pullbacks and universal catenarity

Pullbacks and universal catenarity

Absolutely S-domains and pseudo-polynomial rings

Universally Catenarian Integral Domains, Strong S-Domains and Semistar Operations

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