Chains Of Prime Ideals Research Articles

Chains of prime ideals in power series rings

An ideal I of a commutative ring D with identity is called an SFT ideal if there exist a finitely generated ideal J with J ⊆ I and a positive integer k such that a k ∈ J for each a ∈ I . We prove that for a non-SFT maximal ideal M of an integral domain D , ht ( M 〚 X 〛 / M D 〚 X 〛 ) ≥ 2 ℵ 1 if either (1) D is a 1-dimensional quasi-local domain (in particular D is a 1-dimensional nondiscrete valuation domain) or (2) M is the radical of a countably generated ideal. In other words, if one of the conditions (1) and (2) is satisfied, then there is a chain of prime ideals in D 〚 X 〛 with length at least 2 ℵ 1 such that each prime ideal in the chain lies between M D 〚 X 〛 and M 〚 X 〛 . As an application, assuming the continuum hypothesis we show that if D is either the ring of algebraic integers or the ring of integer-valued polynomials on Z , then dim ⁡ D 〚 X 〛 = ht M 〚 X 〛 = ht ( M 〚 X 〛 / M D 〚 X 〛 ) = 2 ℵ 1 for every maximal ideal M of D .

Growth, relations and prime spectrum of monomial algebras

We show that monomial algebras defined by r_n relations of each degree can (under mild assumptions) be mapped onto prime monomial algebras with prescribed growth functions tightly bounded by any function growing slower than r_n . Using the same method, we are able to construct finitely generated, prime monomial algebras with GK-dimension 2 admitting arbitrarily long chains of prime ideals, providing an answer to a question of Bergman from 1989. Unlike the previously known examples, our algebras are computable, concretely constructed and concrete upper bounds of the form n^2f(n) can be put on their growth rates, for arbitrarily slow unbounded non-decreasing function f .

Krull dimension of power series rings over non-SFT domains

A ring D is called an SFT ring if for each ideal I of D, there exist a finitely generated ideal J of D with J⊆I and a positive integer k such that ak∈J for all a∈I. For a cardinal number α and a ring D, we say that dim⁡D≥α if D has a chain of prime ideals with length ≥α. Arnold showed that if D is a non-SFT ring, then dim⁡D〚X〛≥ℵ0. Let C be the class of non-SFT domains. The class C includes the class of finite-dimensional nondiscrete valuation domains, the class of non-Noetherian almost Dedekind domains, the class of completely integrally closed domains that are not Krull domains, the class of integral domains with non-Noetherian prime spectrum, and the class of integral domains with a nonzero proper idempotent ideal. The ring of algebraic integers, the ring of integer-valued polynomials on Z, and the ring of entire functions are also members of the class C. In this paper we prove that dim⁡D〚X〛≥2ℵ1 for every D∈C and that under the continuum hypothesis 2ℵ1 is the greatest lower bound of dim⁡D〚X〛 for D∈C. On the other hand, there exists a (finite-dimensional) SFT domain D such that dim⁡D〚X〛≥2ℵ1.

Maximal covers of chains of prime ideals

Suppose \(f:S \rightarrow R\) is a ring homomorphism, with S and R not necessarily commutative and f not necessarily unitary. We study the connections between chains in \(\text {Spec} (S)\) and \(\text {Spec} (R)\). We focus on the properties lying over, incomparability, going down (GD), going up (GU) and strong going between (SGB). We prove necessary and sufficient conditions for f to satisfy each of the properties GD, GU and SGB, in terms of maximal \(\mathcal D\)-chains, where \({\mathcal {D}} \subseteq \text {Spec} (S)\) is a nonempty chain. We show that if f satisfies all of the above properties, then every maximal \(\mathcal D\)-chain is a maximal 1:1 cover of \(\mathcal D\). We also present an important example from quasi-valuation theory in which all of the above properties are satisfied. Moreover, we give equivalent conditions for the following property: for every chain \(\mathcal D \subseteq \text {Spec} (S)\) and for every maximal \(\mathcal D\)-chain \(\mathcal C \subseteq \text {Spec} (R)\), \(\mathcal C\) and \(\mathcal D\) are of the same cardinality.

Chains of Prime Ideals and Primitivity of ℤ $\mathbb {Z}$ -Graded Algebras

In this paper we provide some results regarding affine, prime, \(\mathbb {Z}\)-graded algebras \(R=\bigoplus _{i\in \mathbb {Z}}R_{i}\) generated by elements with degrees 1,−1 and 0, with R0 finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If Rk≠0 for almost all k then R is semiprimitive. If in addition R has GK dimension less than 3 then R is either primitive or PI. The tensor product of an arbitrary Brown-McCoy radical algebra of Gelfand Kirillov dimension less than three and any other algebra is Brown-McCoy radical.

Valuation overrings of a Noetherian domain

Let R be a Noetherian domain and 0=P0⊊P1⊊⋯⊊Pn be a saturated chain of prime ideals of R. Let V be a valuation overring of R that has a chain of prime ideals {Qα}α∈Λ such that {Qα∩R}α∈Λ={Pi}i=0n. In this paper, we prove that {Qα}α∈Λ={0=Q0⊊Q1⊊⋯⊊Qn} and VQn is discrete, i.e., QiVQi is principal for all i=1,…,n. Let D be an integral domain such that DP is Noetherian for each prime ideal P of D with htP<∞. As a corollary, we show that if {Pk} is a chain of prime ideals of D such that htPk<∞ for each k, then there exists a discrete valuation overring of D which has a chain of prime ideals lying over {Pk}.

Krull dimension for differential graded algebras

We introduce a naive notion of a system of parameters for a homologically finite complex over a commutative noetherian local ring and compare it to the system of parameters defined by Christensen. We show that these notions differ in general but that they agree when the complex in question is a DG R-algebra. In this case we also show that the Krull dimension defined in terms of the lengths of such systems of parameters agrees with Krull dimensions defined in terms of certain chains of prime ideals.

A generalized principal ideal theorem

The main theorem of this article is an extension of the generalized principal ideal theorem for ideals in Noetherian rings. Instead of requiring the rings to be Noetherian, some natural requirements are imposed on the chains of prime ideals under consideration. The standard (Noetherian) version of the generalized principal ideal theorem is deduced as a corollary and two other applications are presented.

Chains of prime ideals in tensor products of algebras

This paper investigates the length of particular chains of prime ideals in tensor products of algebras over a field k . As an application, we compute dim ( A ⊗ k A ) for a new family of domains A that are k -algebras.

When Are There Infinitely Many Irreducible Elements in a Principal Ideal Domain?

It has been a well-known fact since Euclid's time that there exist infinitely many rational primes. Two natural questions arise: In which other rings, sufficiently similar to the integers, are there infinitely many irreducible elements? Is there a unifying algebraic concept that characterizes such rings? The purpose of the present note is to place the fact concerning the infinity of primes into a more general context, one that also includes the interesting case of the factorial domain (unique factorization domain) of algebraic integers in a number field. We show that, if A is a Principal Ideal Domain (PID, for short), then the fact that A contains infinitely many (pairwise nonassociate) irreducible elements is equivalent to the property that every maximal ideal in the polynomial ring A [x] has the same (maximal) height. We begin by recalling some basic definitions. (We assume that all rings are commutative with an identity element, denoted by 1.) The Jacobson radical J (R) of a ring R is the intersection of all the maximal ideals of R, while the nilradical /O of R is the intersection of all the prime ideals of R. The latter can also be described as the set of all nilpotent elements of R (see [1, Proposition 1.8, p. 5]). The height of a prime ideal P in R is the supremum of the lengths of the chains Po C P1 C ... C Pr = P of prime ideals of R that end at P. The Krull-dimension dim R of R is the supremum of the lengths of all the chains of prime ideals of R, or, equivalently, the supremum of the heights of all the prime ideals P in R. For instance, a field has Krull-dimension 0, while a PID A has Krull-dimension 1, since every prime ideal of A different from (0) has height 1. Finally, two elements a and b of R are associates if there exists a unit u in R such that a = ub. Otherwise, we say that a and b are nonassociates.

On the Prime Ideals in a Commutative Ring

AbstractIf n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

Lifting chains of prime ideals to paravaluation rings

SoitR ⊂T une extension des anneaux commutatifs et soit {Pα :α ∈I} une chaine croissante des ideaux premiers deR (I etant un ensemble totalement ordonne, peut-etre infini). Alors il existe un anneau de paravaluationV deT et une chaine {Qα} des ideaux premiers deV de sorte queR ⊂V etQα ∩R =Pα pour toutα ∈I. Tout d’abord, on etablit le cas special dans lequelT est un corps; dans ce cas, on trouve en effet un tel anneau de valuationV deT. Ensuite, l’assertion ci-dessus pour le cas general decoule comme consequence. Dans le cas general, on peut aussi remplacer le mot “paravaluation” avec le mot “valuation” siR est un anneau de Marot etT est son anneau total de fractions.

Corrigendum: a going-up theorem for arbitrary chains of prime ideals

Corrigendum: a going-up theorem for arbitrary chains of prime ideals

A going-up theorem for arbitrary chains of prime ideals

We prove that if an extension R ⊆ T of commutative rings satisfies the going-up property (for instance, if T is an integral extension of R), then any increasing chain of prime ideals of R (indexed by an arbitrary linearly ordered set) is covered by some corresponding chain of prime ideals of T. As a corollary, we recover the recent result of Kang and Oh that any such chain of prime ideals of an integral domain D is covered by a corresponding chain in some valuation overring of D.

Actions of divided power Hopf algebras

We study prime ideals in enveloping algebra smash products and use a duality construction to obtain results on prime ideals in rings on which divided power Hopf algebras act. These actions correspond to higher derivations. First, we consider chains of prime ideals in an enveloping algebra smash product over an arbitrary ring, where the Lie algebra is assumed to be finite dimensional abelian over a field of positive characteristic. We give a bound on the length of such a chain where the ideals all have the same intersection with the coefficient ring. Then using an explicit construction of a duality theorem of Blattner and Montgomery in this context, we are able to apply results on enveloping algebra smash products to study the invariant ideals of prime ideals in a ring, under a locally nilpotent divided power Hopf algebra action.

Closed and prime ideals in the algebra of bounded analytic functions

Let H∞ be the Banach algebra of all bounded analytic functions in the unit disc. We present a complete description of the closed primary (respectively prime) ideals contained in a maximal ideal of the Shilov boundary of H∞. The paper is also concerned with chains of prime ideals in H∞.

Some curiosities of rings of analytic functions

1. It is the purpose of this paper to exhibit some properties of certain rings of analytic functions which may be a little unexpected. Let E be the ring of all entire functions in one complex variable, i.e. the subring of C[[X]] consisting of all formal power series with infinite convergence radius. More generally, for a subfield K of C let E(K) be the subring of K[[X]] formed by all power series with infinite convergence radius. If Q is a positive real number, let E(Q, K) , resp. J?(Q, K) be the subring of K[[X]] consisting of all power series with convergence radius >Q, resp. 2~. It is well known that E(K) is a non-Noetherian domain and that E(K) is a Bezout domain, i.e. any finitely generated ideal is principal. If K=C this was proved by Wedderburn [ 131 and in the general case by Helmer [4], (who apparently was unaware of [13]). As for the Krull dimension of E the first ‘result’ appeared in [lo] stating that K-dim E = 1. An error in the proof was noticed by Kaplansky [5] and K-dim E is actually infinite. We shall give more precise results concerning the length of chains of prime ideals of E. As shown in [7] the global dimension of E is 23, while the exact value of gl.dimE cannot be determined from the usual axioms of set theory (ZFC): For any t, 3 5 tr m, the statement gl.dim E = t is consistent with ZFC, in fact, even consistent with ZFC + MA, (MA denoting Martin’s axiom). The corresponding results hold true if E is replaced by E(K) or &, K). The proofs only require minor modifications. For E(Q, K), however, the situation is completely different. For any positive Q and any field KS C the ring E(Q, K) is Euclidean, in particular a PID. Since for instance E(1, C) = n;=, E(l l/n, 4X) we obtain a decreasing sequence of PID’s whose intersection is a Bezout domain of undecidable global dimension and of uncountable Krull dimension. The stable range (in the sense of Bass [l]) of the above rings depends on K. If KS R the stable range of each of the rings E(Q, K), E(Q, K) and E(K) is 2, otherwise, when Kg R the stable range is 1.

Chain of Prime Ideals in Formal Power Series Rings

Chain of Prime Ideals in Formal Power Series Rings

Proper morphisms and excellent schemes

Let f: X → Y be a finite type morphism of locally noetherian schemes. It is well known ([3, IV, 7.8.6]) that the excellent property ascends from Y to X. On the other side there are counter-examples where X is excellent and Y is not. First of all it is easy to show that the condition on chains of prime ideals does not descend (see [3, IV, 7.8.4]), even by finite morphisms. Secondly in [2] it is produced an example where X is excellent while Y is not a G-scheme (i.e. it has not the good properties of formal fibers). However in [2] it is also proved that the property concerning the openness of regular loci (the so called “J-2”) descends by finite type surjective morphisms. Therefore we are led to the following question: When does the G-scheme property descend? I.e. what conditions do we need on f? A reasonable condition is conjectured (in [2]) as the following: f is proper surjective. The aim of the present paper is precisely to give an answer to such a question. What we really prove is the following. If X is a G-scheme and J-2 (quasi-excellent), then the same is true for Y, provided that f is proper surjective and moreover all the residue fields of Y have characteristic 0. We remark that the result is strongly based on Hironaka’s desingularization for quasi-excellent schemes defined over a field of characteristic 0.

Chains of Prime Ideals in Enveloping Algebras of Solvable Lie Algebras

Chains of Prime Ideals in Enveloping Algebras of Solvable Lie Algebras