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Let $P$ be a finitely generated ideal of a commutative ring $R$. Krull's Principal Ideal Theorem states that if $R$ is Noetherian and $P$ is minimal over a principal ideal of $R$, then $P$ has height at most one. Straightforward examples show that this assertion fails if $R$ is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem., 15 pages, to appear in Math. Proc. Cambridge Philos. Soc

We prove that the Krull dimension of the ring of holomorphic functions of a connected complex manifold is at least continuum if it is positive., Comment: 6 pages. An error pointed out by Pete Clark is corrected. The stronger statement about the Krull dimension at least continuum is proven

for example, because of their connections with number theory and arithmetic geometry; see the paper by Coates, Schneider and Sujatha [4] for more details. When G is a uniform pro-p group, ΛG is a concrete example of a complete local Noetherian ring (noncommutative, in general) with good homological properties: it is known that ΛG has finite global dimension and is an Auslander regular ring. Thus, ΛG falls into the class of rings studied by Brown, Hajarnavis and MacEacharn in [1]. There they consider various properties of Noetherian rings R of finite global dimension, including the Krull(–Gabriel–Rentschler) dimension K(R)—a module-theoretic dimension which measures how far R is from being Artinian. They also posed the following question

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We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the supremum of the length of independent sequences. The proof has led to other notions of independent sequences, which have interesting applications. For example, we can characterize the maximum number of analytically independent elements in an arbitrary ideal of a local ring and that dim B is not greater than dim A if B is a subalgebra of A and A is a (not necessarily finitely generated) subalgebra of a finitely generated algebra over a Noetherian Jacobson ring., This is a revised version of the submitted manuscript

Let R be a commutative Noetherian ring with non-zero identity and a be a maximal ideal of R. An R-module M is called minimax if there is a finitely generated submodule N of M such that M/N is Artinian. Over a Gorenstein local ring R of finite Krull dimension, we proved that the Socle of H (a) (n) (R) is a minimax R-module for each n a parts per thousand yen 0.

In this paper, we are concerned with the study of the dimension theory of tensor products of algebras over a field $k$. We introduce and investigate the notion of generalized AF-domain (GAF-domain for short) and prove that any $k$-algebra $A$ such that the polynomial ring in one variable $A[X]$ is an AF-domain is in fact a GAF-domain, in particular any AF-domain is a GAF-domain. Moreover, we compute the Krull dimension of $A\otimes_kB$ for any $k$-algebra $A$ such that $A[X]$ is an AF-domain and any $k$-algebra $B$ generalizing the main theorem of Wadsworth in [16]., 15 pages

We study the "q-commutative" power series ring R:=k_q[[x_1,...,x_n]], defined by the relations x_ix_j = q_{ij}x_j x_i, for multiplicatively antisymmetric scalars q_{ij} in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In particular, we prove that the prime spectrum of R is normally separated and is finitely stratified by commutative noetherian spectra. Combining this normal separation with results of Chan, Wu, Yekutieli, and Zhang, we are able to conclude that R is catenary. Following the approach of Brown and Goodearl, we also show that links between prime ideals are provided by canonical automorphisms. Moreover, for sufficiently generic q_{ij}, we find that R has only finitely many prime ideals and is a UFD (in the sense of Chatters)., Revised; to appear in Algebras and Representation Theory. In this (final) version, catenarity is established, by combining normal separation with results of Chan, Wu, Yekutieli, and Zhang.

Let N be a finitely generated module over a Noetherian local ring (R,m). We give criteria for the height of the order ideal N^*(x) of an element x \in N to be bounded by the rank of N. The Generalized Principal Ideal Theorem of Bruns, Eisenbud and Evans says that this inequality always holds if x \in mN, but if x is a minimal generator the question is more subtle; for example, when R is graded and N represents a vector bundle on Proj R, the question is connected with the vanishing of the top chern class. We show that the inequality does hold if the hypothesis x \in mN becomes true after first extending scalars to some local domain and then factoring out torsion. We also give conditions under which it holds in terms of residual intersections and integral closures of modules. We derive information about order ideals that leads to bounds on the heights of trace ideals of modules--even in circumstances where we do not have the expected bounds for the heights of the order ideals., 20 pages

The results here generalise [2, Proposition 4.3] and [9, Theorem 5.11]. We shall prove the following.THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotent prime ideal of R such that PRis projective. Then P is left localisable and RPis a prime principal left and right ideal ring.We also have the following theorem.THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotent maximal ideal of R such that MRis projective. Then M has the left AR-property and M contains a right regular element of R.

The main aim of this paper is to show that an AB5*-module whose small submodules have Krull dimension has a radical having Krull dimension. The proof uses the notion of dual Goldie dimension., to appear in the Miskolc conference proceeding 1996

Let k be an algebraically closed field of characteristi c zero, and A a finitely generated /c-algebra of Krull dimension at most one. In this paper we study the ring of differential operators 3ί{A). For example we obtain necessary and sufficient conditions for 2$ (A) to be a direct sum of Simple rings, or to be left or right Noetherian. 0.1. Let k be an algebraically closed field of characteristi c zero and A a finitely generated (commutative) /c-algebra. The primary purpose of this paper is to study the ring 3f[A) of differential operators on A when dim(^4), the Krull dimension of A, is at most one. If A is also reduced or is a domain 3f(A) has been studied extensively in [10] and [15] and we prove analogues of the main results of these papers. For example THEOREM A. Let A be a finitely generated k-algebra with Krull

In this paper we prove that for a commutative noetherian ring R with Krull dim R = d < ∞ and a submonoid M of Q′+ for some rεN such that M⊂Q′+ is an integral extension, the group of elementary matrices En(R[M]) acts transitively on Umn[R[M]) for all n ⩾ max(d + 2, 3). Hitherto the special case M=Q′+ was obtained in [Su]. Bass-Vasershtein's classical results imply the transitivity only for n ⩾ d + 2 + r. The key step is the consideration of finitely generated normal monoids. The proof we give here uses some density properties of integer lattices.

Commutative monoids yield an analogy between the theory of factorization in commutative integral domains and the theory of direct sum decompositions of modules. We show that the monoid V( C ) of isomorphism classes of a class C of modules with semilocal endomorphism rings is a Krull monoid (Theorem 3.4). Krull monoids often appear in the study of factorizations of elements in integral domains, and are defined as the monoids V for which there is a divisor homomorphism of V into a free commutative monoid. In particular, we consider the case in which C is the class of biuniform modules. For this class the validity of a weak form of the Krull–Schmidt Theorem is explained via a representation of V( C ) as a subdirect product of free commutative monoids.

We obtain formulae for the Krull dimension of the generalized Weyl algebra T = R(σ, a), where R is a left noetherian ring.

We prove that ifA→G→Qis a short exact sequence of groups whereGis finitely generated,AandQare abelian,Ais a2-torsion Krull dimension one3Q-module via conjugation then theFPm-Conjecture holds. In general ifGis of typeFPmand either the extension is split orAis4-torsion we show thatAism-tame as5Q-module.

Let D be a Krull domain with quotient field K. We study the class group of the integer-valued polynomial ring over D, Int(D)≔{f∈K[X];f(D)⊆D}. In particular, we give necessary and sufficient conditions on D for the class group of Int(D) to be generated by the classes of the t-invertible t-prime ideals and, in this case, we describe its generators. A case of particular interest is when D is a UFD. We also characterize Krull domains D for which Int(D) is a GCD-domain.

Many rings that have enjoyed growing interest in recent years, e.g., quantum enveloping algebras, quantum matrices, certain Witten-algebras,…,can be presented as generalized Weyl algebras. In the paper we develop techniques for calculating dimensions, here mainly the Krull dimension in the sense of Rentschler–Gabriel, of such generalized Weyl algebras and specify the results for some popular algebras including those mentioned above.

Let A be a subring of the integral domain B . Then B is said to be t-linked over A if for each finitely generated ideal I of A with I -1 = A , we have ( IB ) -1 = B . If A and B are Krull domains, this condition is equivalent to PDE. We show that if B is t-linked over A , then the map I →( IB ) t gives a homomorphism from the group of t-invertible t-ideals of A to the group of t-invertible t-ideals of B and hence a homomorphism Cl t ( A )→Cl t ( B ) of the t-class groups. Conditions are given for these maps to be surjective which extend Nagata's Theorem for Krull domains to a much larger class of domains including, e.g., Noetherian domains each of whose grade-one prime ideals has height one.

5.1. A Krull pair (E, R) is called strong if every maximal element F in the Asano class 01 of E is d-Noetherian and R is a left-Asano quotient ring of E. This condition entails that every thick divisorial left ideal of every maximal order in iy is symbolically invertible, that is, a member of the Brandt groupoid B&Q) = K (see Theorem 3.6 and Corollary 3.8). In Section 6, the groupoid k’ is investigated from a general point of view. This is used in Section 7 to obtain a factorization theory for one-sided ideals of a strong Krull pair. But since we now have completed the essential definitions, it is appropriate first to point out examples of Krull pairs and strong Krull pairs showing that important classes of rings are of the type considered in this paper.

We prove the existence of universal spaces for the Krull dimension. To prove it, we must characterize when a continuous map between finite spaces factors through a finite space of dimension ⩽ n . Moreover, we deduce some consequences for the (Krull) dimension of distributive lattices.

Let D ⊆ E be an extension of integral domains, Γ be a nonzero torsion-free (additive) grading monoid with quotient group G such that Γ ∩ − Γ = { 0 } . Set Γ ⁎ = Γ ∖ { 0 } and R = D + E [ Γ ⁎ ] . In this paper, we show that if G satisfies the ascending chain condition on cyclic subgroups, then R is a generalized Krull domain (resp., generalized unique factorization domain) if and only if D = E , D is a generalized Krull domain (resp., generalized unique factorization domain) and Γ is a generalized Krull semigroup (resp., weakly factorial GCD-semigroup).

We give a constructive deciphering for a generalization of the Quillen–Suslin theorem due to Maroscia and Brewer & Costa stating that finitely generated projective modules over R[X1,…,Xn], where R is a Prüfer domain with Krull dimension ≤1, are extended from R.

We show that a ring is a Krull ring if and only if every nonzero regular prime ideal contains a t-invertible prime ideal if and only if every proper regular principal ideal is quasi-equal to a product of prime ideals.

The principal ideal theorem is one of the basic results in the theory of commutative Noetherian rings. A. W. Chattess [l] and P. F. Smith [6] have obtained an analogue of it for those noncommutative Noetherian rings in which the localization at an arbitrary prime ideal can be performed in the usual way. The object of this note is to prove an analogue of the principal ideal theorem for prime Noetherian rings satisfying some polynomial identity. A very neat proof of the principal ideal theorem for commutative Noetherian rings is given by Kaplansky [4, p. 1041. Th e o b vious obstacle in adopting it to the noncommutative situation is that the proof uses the usual localization at a prime ideal whereas no sufficiently nice localization is available at an arbitrary prime ideal in a prime Noetherian ring satisfying a polynomial identity. A closer examination of Kaplansky’s proof reveals that it really hinges on the additivity of the notion of length derived from the JordanHolder theorem; localization is used only to produce a situation in which the Jordan-Holder theorem becomes applicable. We have established in [3] a version of the Jordan-Holder theorem which is applicable to an arbitrary finitely generated module over an FBN-ring (see definition below). It leads to a notion of length which has enough additivity (Theorem 1). Using this notion of length in place of the usual one, we have been able to recast Kaplansky’s proof in such a way that it works for prime Noetherian rings satisfying some polynomial identity.

Texto completo. Acesso restrito. p. 63 - 78 Submitted by Santiago Fabio (fabio.ssantiago@hotmail.com) on 2013-02-05T16:34:27Z No. of bitstreams: 1 111.pdf: 691571 bytes, checksum: 96d4a77cafdb17246efdc275a4ee2b8d (MD5) Made available in DSpace on 2013-02-05T16:34:27Z (GMT). No. of bitstreams: 1 111.pdf: 691571 bytes, checksum: 96d4a77cafdb17246efdc275a4ee2b8d (MD5) Previous issue date: 1988 For a finitely generated module E over the Noetherian ring R we consider formulas for the Krull dimension of the symmetric algebra S(E). A result of Huneke and Rossi is re-proved and an eiffective formulais derived that reads the dimension of S(E) from a presentation of E. They provide a first line of obstructions for S(E) to be an integral domain. For algebras of codimension at most four we give methods, including computer-assisted ones, to ascertain whether S(E) is a Cohen-Macaulay domain. Salvador

In a previous paper on this subject, the author gave a new proof of the theorem of "analytic independence of systems of parameters". The methods used can be applied to prove a converse result. Hence: An ideal of a Noetherian ring is of the principal class if, and only if, it is generated by an analytically independent set.

Let T be the set of minimal primes of a Krull domain A. If S is a subset of T9 we form B = n AP for PeS and study the relation of the class group of B to that of A. We find that the class group of B is always a homomorphic image of that of A. We use this type of construction to obtain a Krull domain with specified class group and then alter such a Krull domain to obtain a Dedekind domain with the same class group. Let A be a Krull domain with quotient field K. Thus A is an intersection of rank 1 discrete valuation rings; and if x e K, x is a unit in all but a finite number of these valuation rings. If P is a minimal prime ideal of A, then AP is a rank 1 discrete valuation ring and must occur in any intersection displaying A as a Krull domain. In fact, if T denotes the set of minimal prime ideals of A, then A = OPΘΓAP displays A as a Krull domain. Choose a subset S of T (S Φ 0) and form the domain B — Γ\pesAp. It is immediate that B is also a Krull domain which contains A and has quotient field K. If one of the AP were eliminable from the intersection representing B, it would also be eliminable from that representing A. Thus the AP for PeS are exactly the rings of the type BQ, where Q is a minimal prime ideal of B. If Q is minimal prime ideal of B, then Q Π A = P for the Pe S such that BQ = AP. Let A and B be generic labels throughout this paper for a Krull domain A and a Krull domain B formed from A as above. We recall that the valuation rings AP are called the essential valuation rings, and we will denote by VP the valuation of A going with AP. We summarize and add a complement to the above. PROPOSITION 1. With A and B as above, B is a Krull domain containing A, and the AP for PeS are the essential valuation rings of JB. Every ring B is of the form AM for some multiplicative set M if and only if the class group of A is torsion.