On the Prime Spectrum of an Integral Domain with Finite Spectral Semistar Operations

2011 ◽  
Vol 18 (spec01) ◽  
pp. 965-972 ◽  
Author(s):  
A. Mimouni
Keyword(s):  
Finite Number ◽  
Integral Domain ◽  
Krull Dimension ◽  
Prime Spectrum

In this paper, we investigate the prime spectrum of an integral domain R with a finite number of spectral semistar operations. This will be done by seeking for a possible link between the cardinality of the set SpSS (R) of all spectral semistar operations on R and its Krull dimension. In particular, we prove that if | SpSS (R)|=n+ dim R, then 2| Max (R)|≤ n+1. This leads us to give a complete description for the spectrum of a domain R such that | SpSS (R)|=n+ dim R for 1 ≤ n ≤ 5.

INTEGRAL DOMAINS OF THE FORM A + XI[[X]]: PRIME SPECTRUM, KRULL DIMENSION

2005 ◽  
Vol 04 (06) ◽  
pp. 599-611
Author(s):  
SANA HIZEM ◽  
ALI BENHISSI
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Integral Domain ◽  
Finite Dimension ◽  
Krull Dimension ◽  
Necessary Condition ◽  
Prime Spectrum ◽  
Prüfer Domain ◽  
Integral Domains ◽  
Contraction Map

Let A be an integral domain, X an analytic indeterminate over A and I a proper ideal (not necessarily prime) of A. In this paper, we study the ring [Formula: see text] First, we study the prime spectrum of R. We prove that the contraction map: Spec (A[[X]]) → Spec (R); Q ↦ Q ∩ R induces a homeomorphism, for the Zariski's topologies, from {Q ∈ Spec (A[[X]]) | XI[[X]] ⊈ Q} onto {P ∈ Spec (R) | XI[[X]] ⊈ P}. If P ∈ Spec (R) is such that XI[[X]] ⊆ P then there exists p ∈ Spec (A) such that P = p + XI[[X]]. Next, we study the Krull dimension of R. We give a necessary condition for R to be of finite Krull dimension. In particular, if R is of finite dimension then I must be an SFT ideal of A. Then we determine bounds for dim (R). Examples are given to indicate the sharpness of the results. In case I is a maximal ideal of A and A is either a Noetherian ring, SFT Prüfer domain or A[[X]] is catenarian and I SFT, we establish that dim (R) = dim (A[[X]]) = dim (A) + 1. Finally, we examine the possible transfer of the LFD property and the catenarity between the rings A, A[[X]] and R in case I is a maximal ideal of A.

scholarly journals A Remark on Coherent Overrings

1978 ◽  
Vol 21 (3) ◽  
pp. 373-375 ◽  
Author(s):  
Ira J. Papick
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
General Theorem ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Quotient Field ◽  
Valuation Rings ◽  
Finitely Presented

Throughout this note, let R be a (commutative integral) domain with quotient field K. A domain S satisfying R ⊆ S ⊆ K is called an overring of R, and by dimension of a ring we mean Krull dimension. Recall [1] that a commutative ring is said to be coherent if each finitely generated ideal is finitely presented.In [2], as a corollary of a more general theorem, Davis showed that if each overring of a domain R is Noetherian, then the dimension of R is at most 1. (This corollary is the converse of a version of the Krull-Akizuki Theorem [5, Theorem 93], and can also be proved directly by using the existence of valuation rings dominating finite chains of prime ideals [4, Corollary 16.6].) It is our purpose to prove that if R is Noetherian and each overring of R is coherent, then the dimension of £ is at most 1. We shall also indicate some related questions and examples.

Primary Ideals and Prime Power Ideals

1966 ◽  
Vol 18 ◽  
pp. 1183-1195 ◽  
Author(s):  
H. S. Butts ◽  
Robert W. Gilmer
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Prime Power ◽  
Ideal Theory ◽  
Dedekind Domain ◽  
Primary Ideal ◽  
Primary Ideals ◽  
The Ideal

This paper is concerned with the ideal theory of a commutative ringR.We sayRhas Property (α) if each primary ideal inRis a power of its (prime) radical;Ris said to have Property (δ) provided every ideal inRis an intersection of a finite number of prime power ideals. In (2, Theorem 8, p. 33) it is shown that ifDis a Noetherian integral domain with identity and if there are no ideals properly between any maximal ideal and its square, thenDis a Dedekind domain. It follows from this that ifDhas Property (α) and is Noetherian (in which caseDhas Property (δ)), thenDis Dedekind.

Krull Dimension of Injective Modules Over Commutative Noetherian Rings

2005 ◽  
Vol 48 (2) ◽  
pp. 275-282
Author(s):  
Patrick F. Smith
Keyword(s):  
Noetherian Ring ◽  
Integral Domain ◽  
Krull Dimension ◽  
Injective Module ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
One Dimensional ◽  
Injective Modules

AbstractLet R be a commutative Noetherian integral domain with field of fractions Q. Generalizing a forty-year-old theorem of E. Matlis, we prove that the R-module Q/R (or Q) has Krull dimension if and only if R is semilocal and one-dimensional. Moreover, if X is an injective module over a commutative Noetherian ring such that X has Krull dimension, then the Krull dimension of X is at most 1.

CPI-Extensions: Overrings of Integral Domains with Special Prime Spectrums

1977 ◽  
Vol 29 (4) ◽  
pp. 722-737 ◽  
Author(s):  
Monte B. Boisen ◽  
Philip B. Sheldon
Keyword(s):  
Topological Space ◽  
Partially Ordered Set ◽  
Integral Domain ◽  
Ordered Set ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Integral Domains ◽  
Partially Ordered ◽  
Set Inclusion

Throughout this paper the term ring will denote a commutative ring with unity and the term integral domain will denote a ring having no nonzero divisors of zero. The set of all prime ideals of a ring R can be viewed as a topological space, called the prime spectrum of R, and abbreviated Spec (R), where the topology used is the Zariski topology [1, Definition 4, § 4.3, p. 99]. The set of all prime ideals of R can also be viewed simply as aposet - that is, a partially ordered set - with respect to set inclusion. We will use the phrase the pospec of R, or just Pospec (/v), to refer to this partially ordered set.

On the associativity of the torsion functor

1974 ◽  
Vol 10 (1) ◽  
pp. 107-118
Author(s):  
John Clark
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Noetherian Ring ◽  
Integral Domain ◽  
Direct Sum ◽  
Torsion Functor

Let R be a commutative ring with identity. We say that tor is associative over R if for all R-modules A, B, C there is an isomorphism Our main results are that (1) tor is associative over a noetherian ring R if and only if R is the direct sum of a finite number of Dedekind rings and uniserial rings, and (2) tor is associative over an integral domain R if and only if R is a Prüfer ring.

scholarly journals Primitiewe elemente vir kommutatiewe ringuitbreidings

1991 ◽  
Vol 10 (2) ◽  
pp. 67-71
Author(s):  
H. J. Schutte
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Integral Domain ◽  
Prime Ideals ◽  
Primitive Elements ◽  
Integral Domains ◽  
Field Extensions ◽  
Ring Extensions ◽  
Minimal Prime

The existence of primitive elements for integral domain extensions is considered with reference to the well known theorem about primitive elements for field extensions. Primitive elements for extensions of a commutative ring R with identity are considered, where R has only a finite number of minimal prime ideals with zero intersection. This case is reduced to the case for ring extensions of integral domains.

Nagata-like theorems for integral domains of finite character and finite t-character

2015 ◽  
Vol 14 (08) ◽  
pp. 1550119
Author(s):  
D. D. Anderson ◽  
Gyu Whan Chang ◽  
Muhammad Zafrullah
Keyword(s):  
Finite Number ◽  
Integral Domain ◽  
Integral Domains ◽  
Maximal Ideals

An integral domain D is said to be of finite character (resp., finite t-character) if every nonzero nonunit of D belongs to at most a finite number of maximal ideals (resp., maximal t-ideals) of D. Let S be a multiplicative set of D. In this paper we study when DS being of finite character (resp., finite t-character) implies that D is of finite character (resp., finite t-character).

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