On the krull and valuative dimension of D + XDs[X] domains

Marco Fontana; Salah Kabbaj

doi:10.1016/0022-4049(90)90069-T

On the krull and valuative dimension of D + XD_s[X] domains

Marco Fontana^*, Salah Kabbaj

^*Corresponding author for this work

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

16 Scopus citations

Abstract

In this paper, we deal with the integral domain D(^S,r):=D+(X₁,X₂,...,X_r)D_S[X₁, X₂,...,X_r], where D is an integral domain and S is a multiplicative set of D. The purpose is to pursue the study, initiated by Costa-Mott-Zafrullah in 1978, concerning the prime ideal structure of such domains. We characterize when D(^S,r) is a strong S-domain, a stably strong S-domain, a catenarian domain and a universally catenarian domain. As a consequence, we obtain a new class of non-Noetherian universally catenarian domains. Moreover, we give an explicit formula for the Krull dimension of D(^S,r) (depending on S and on the Krull dimensions of D and D_S[X₁,X₂,...,X_r]) and we compute its valuative dimension.

Original language	English
Pages (from-to)	231-245
Number of pages	15
Journal	Journal of Pure and Applied Algebra
Volume	63
Issue number	3
DOIs	https://doi.org/10.1016/0022-4049(90)90069-T
State	Published - 10 Apr 1990
Externally published	Yes

Bibliographical note

Funding Information:
The authors gratefully acknowledge partial support from NATO Collaborative Research Grant RG 85/0035. This work was partially prepared at the University of Lyon I, during the visit of Fontana, summer 1988. The final version of the paper has greatly benefited by some useful comments of the referee.

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1016/0022-4049(90)90069-T

Cite this

@article{765e327b8e954358bd2966835b142c0d,

title = "On the krull and valuative dimension of D + XDs[X] domains",

abstract = "In this paper, we deal with the integral domain D(S,r):=D+(X1,X2,...,Xr)DS[X1, X2,...,Xr], where D is an integral domain and S is a multiplicative set of D. The purpose is to pursue the study, initiated by Costa-Mott-Zafrullah in 1978, concerning the prime ideal structure of such domains. We characterize when D(S,r) is a strong S-domain, a stably strong S-domain, a catenarian domain and a universally catenarian domain. As a consequence, we obtain a new class of non-Noetherian universally catenarian domains. Moreover, we give an explicit formula for the Krull dimension of D(S,r) (depending on S and on the Krull dimensions of D and DS[X1,X2,...,Xr]) and we compute its valuative dimension.",

author = "Marco Fontana and Salah Kabbaj",

note = "Funding Information: The authors gratefully acknowledge partial support from NATO Collaborative Research Grant RG 85/0035. This work was partially prepared at the University of Lyon I, during the visit of Fontana, summer 1988. The final version of the paper has greatly benefited by some useful comments of the referee.",

year = "1990",

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T1 - On the krull and valuative dimension of D + XDs[X] domains

AU - Fontana, Marco

AU - Kabbaj, Salah

N1 - Funding Information: The authors gratefully acknowledge partial support from NATO Collaborative Research Grant RG 85/0035. This work was partially prepared at the University of Lyon I, during the visit of Fontana, summer 1988. The final version of the paper has greatly benefited by some useful comments of the referee.

PY - 1990/4/10

Y1 - 1990/4/10

N2 - In this paper, we deal with the integral domain D(S,r):=D+(X1,X2,...,Xr)DS[X1, X2,...,Xr], where D is an integral domain and S is a multiplicative set of D. The purpose is to pursue the study, initiated by Costa-Mott-Zafrullah in 1978, concerning the prime ideal structure of such domains. We characterize when D(S,r) is a strong S-domain, a stably strong S-domain, a catenarian domain and a universally catenarian domain. As a consequence, we obtain a new class of non-Noetherian universally catenarian domains. Moreover, we give an explicit formula for the Krull dimension of D(S,r) (depending on S and on the Krull dimensions of D and DS[X1,X2,...,Xr]) and we compute its valuative dimension.

AB - In this paper, we deal with the integral domain D(S,r):=D+(X1,X2,...,Xr)DS[X1, X2,...,Xr], where D is an integral domain and S is a multiplicative set of D. The purpose is to pursue the study, initiated by Costa-Mott-Zafrullah in 1978, concerning the prime ideal structure of such domains. We characterize when D(S,r) is a strong S-domain, a stably strong S-domain, a catenarian domain and a universally catenarian domain. As a consequence, we obtain a new class of non-Noetherian universally catenarian domains. Moreover, we give an explicit formula for the Krull dimension of D(S,r) (depending on S and on the Krull dimensions of D and DS[X1,X2,...,Xr]) and we compute its valuative dimension.

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