Noetherian Domain Research Articles

Chow groups of one-dimensional noetherian domains

Chow groups of one-dimensional noetherian domains

Integral closure of an affine algebra

Let R be a commutative ring with identity and R ′ be the integral closure of R. In this paper, we show that if R is an affine algebra over a field K, then every regular ideal of R ′ is finitely generated, i.e., R ′ is an r-Noetherian ring. We also study when the integral closure of an affine algebra is Noetherian. First we show that if R is a Krull ring such that R/P is a Noetherian domain for each minimal regular prime ideal P of R, then R is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that if R is an r-Noetherian ring with reg-dim R ≤ 2 , then R ′ is an r-Noetherian ring. We finally construct a couple of r-Noetherian rings, e.g., an r-Noetherian ring R that is not Noetherian and reg-dim R = ∞ or reg-dim R = n ≤ dim R = n + m − 1 for arbitrary positive integers n, m.

Effective generic freeness and applications to local cohomology

AbstractLet be a Noetherian domain and be a finitely generated ‐algebra. We study several features regarding the generic freeness over of an ‐module. For an ideal , we show that the local cohomology modules are generically free over under certain settings where is a smooth ‐algebra. By utilizing the theory of Gröbner bases over arbitrary Noetherian rings, we provide an effective method to b make explicit the generic freeness over of a finitely generated ‐module.

Specht property for the graded identities of the pair (M2(D),sl2(D))

Let D be a Noetherian infinite integral domain, denote by M2(D) and by sl2(D) the 2×2 matrix algebra and the Lie algebra of the traceless matrices in M2(D), respectively. In this paper we study the weak polynomial identities for the natural grading by the cyclic group Z2 of order 2 on M2(D) and on sl2(D). We describe a finite basis of the graded polynomial identities for the pair (M2(D),sl2(D)). Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for (M2(D),sl2(D)), is finitely generated. The polynomial identities for M2(D) are known if D is any field of characteristic different from 2. The identities for the Lie algebra sl2(D) are known when D is an infinite field. The identities for the pair we consider were first described by Razmyslov when D is a field of characteristic 0, and afterwards by the second author when D is an infinite field. The graded identities for the pair (M2(D),gl2(D)) were also described, by Krasilnikov and the second author.In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets.

Partial trace ideals, torsion and canonical module

For any finitely generated module M with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant h(M) was introduced and studied in [25]. We establish a bound on it which helps capture information about the torsion submodule of M when M has rank one and generalizes the discussion in [25]. We further study bounds and properties of h(M) in the case when M is the canonical module ωR. This in turn helps in answering a question of S. Greco and then provides classifications in the Gorenstein, almost Gorenstein and far-flung Gorenstein setups.

A degree bound for rings of arithmetic invariants

Consider a Noetherian domain R and a finite group G⊆Gln(R). We prove that if the ring of invariants R[x1,…,xn]G is a Cohen-Macaulay ring, then it is generated as an R-algebra by elements of degree at most max⁡(|G|,n(|G|−1)). As an intermediate result we also show that if R is a Noetherian local ring with infinite residue field then such a ring of invariants of a finite group G over R contains a homogeneous system of parameters consisting of elements of degree at most |G|.

Every finite poset is isomorphic to a saturated subset of the spectrum of a Noetherian UFD

We show that every finite poset is isomorphic to a saturated subset of the spectrum of a Noetherian unique factorization domain. In addition, we show that every finite poset is isomorphic to a saturated subset of the spectrum of a quasi-excellent domain.

On quasi-morphic modules

An R-module M is called quasi-morphic if for any f∈EndR(M), there exist g,h∈EndR(M) such that Imf=Kerg and Kerf=Imh. In addition, MR is said to be morphic whenever g=h in the above definition. The main objective of this paper is investigating quasi-morphic property for several classes of modules. First we obtain general properties of quasi-morphic modules via exact sequence approach. Moreover, we investigate conditions under which a finite length quasi-morphic module is morphic. As a result, we show that for uniserial finite length modules, the notions of morphic and quasi-morphic coincide. Over a principal ideal domain R, direct sums of cyclic modules which are (quasi-)morphic are characterized. Among applications of our results, nonsingular extending (quasi-)morphic modules are characterized completely. We also prove that over a commutative Noetherian domain R which is not a field, quasi-morphic nonsingular extending modules are precisely direct sums of copies of Q (the quotient field of R). (Quasi-)Morphic singular extending abelian groups are also characterized.

Finite generation of nilpotent quotients of fundamental groups of punctured spectra

In SGA 2, Grothendieck conjectures that the étale fundamental group of the punctured spectrum of a complete noetherian local domain of dimension at least two with algebraically closed residue field is topologically finitely generated. In this paper, we prove a weaker statement, namely that the maximal pro-nilpotent quotient of the fundamental group is topologically finitely generated. The proof uses p-adic nearby cycles and negative definiteness of intersection pairings over resolutions of singularities as well as some analysis of Lie algebras of certain algebraic group structures on deformation cohomology.

Frobenius splitting, strong F-regularity, and small Cohen-Macaulay modules

Let M M be a finitely generated module over an (F-finite local) ring R R of prime characteristic p > 0 p >0 . Let e M {}^e\!M denote the result of restricting scalars using the map F e : R → R F^e\colon R \to R , the e e\, th iteration of the Frobenius endomorphism. Motivated in part by the fact that in certain circumstances the splitting of e M {}^e\!M as e e grows can be used to prove the existence of small (i.e., finitely generated) maximal Cohen-Macaulay modules, we study splitting phenomena for e M {}^e\!M from several points of view. In consequence, we are able to prove new results about when one has such splittings that generalize results previously known only in low dimension, we give new characterizations of when a ring is strongly F-regular, and we are able to prove new results on the existence of small maximal Cohen-Macaulay modules in the multi-graded case. In addition, we study certain corresponding questions when the ring is no longer assumed F-finite and purity is considered in place of splitting. We also answer a question, raised by Datta and Smith, by showing that a regular Noetherian domain, even in dimension 2, need not be very strongly F-regular.

On a weak version of S-Noetherianity

In this paper, we introduce a new class of ring called [Formula: see text]-[Formula: see text]-Noetherian ring, which is a weak version of [Formula: see text]-Noetherian ring property and study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamation of rings. Furthermore, we define the concept of nonnil [Formula: see text]-[Formula: see text]-Noetherian ring property which is a generalization of the [Formula: see text]-[Formula: see text]-Noetherian domain property and establish a characterization of this notion using pullbacks.

Derivations and root closures of graded ideals

We show that the graded structure of an ideal is preserved under root closures. As an application, for a locally nilpotent derivation [Formula: see text] on a Noetherian integral domain [Formula: see text] containing a field of characteristic zero, we show that if [Formula: see text] is [Formula: see text]-invariant, then so are its root closures.

Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings

The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the $S$-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the $S$-variant of Eakin–Nagata–Eisenbud theorem for general rings by using $S$-Noetherian modules. We also show that every right $S$-Noetherian domain is right Ore, which is embedded into a division ring. For a right $S$-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right $S$-Noetherian or right Noetherian. As applications, the $S$-variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.

Harish-Chandra Modules over Hopf Galois Orders

Abstract The theory of Galois orders was introduced by Futorny and Ovsienko [9]. We introduce the notion of $\mathcal {H}$-Galois $\Lambda $-orders. These are certain noncommutative orders $F$ in a smash product of the fraction field of a noetherian integral domain $\Lambda $ by a Hopf algebra ${\mathcal {H}}$ (or, more generally, by a coideal subalgebra of a Hopf algebra). They are generalizations of Webster’s [25] principal flag orders. Examples include Cherednik algebras, as well as examples from Hopf Galois theory. We also define spherical Galois orders, which are the corresponding generalizations of principal Galois orders introduced by the author [12]. The main results are (1) for every maximal ideal $\mathfrak {m}$ of $\Lambda $ of finite codimension, there exists a simple Harish-Chandra $F$-module in the fiber of $\mathfrak {m}$; (2) for every character of $\Lambda $, we construct a canonical simple Harish-Chandra module as a subquotient of the module of local distributions; (3) if a certain stabilizer coalgebra is finite-dimensional, then the corresponding fiber of simple Harish-Chandra modules is finite; and (4) centralizers of symmetrizing idempotents are spherical Galois orders and every spherical Galois order appears that way.

On noncommutative bounded factorization domains and prime rings

A ring has bounded factorizations if every cancellative nonunit a∈R can be written as a product of atoms and there is a bound λ(a) on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations.

Quantum tilting modules over local rings

AbstractWe show that tilting modules for quantum groups over local Noetherian domains of quantum characteristic 0 exist and the indecomposable tilting modules are parametrized by their highest weight. For this, we introduce a model category associated with a Noetherian ‐domain and a root system . We show that if is of quantum characteristic , the model category contains all ‐modules that admit a Weyl filtration. If is in addition local, we study torsion phenomena in the model category. This leads to a construction of torsion free, or “maximal” objects in . We show that these correspond to tilting modules for the quantum group associated with and .

Some improvements for the algorithm of Gröbner bases over dual valuation domain

<abstract><p>As a special ring with zero divisors, the dual noetherian valuation domain has attracted much attention from scholars. This article aims at to improve the Buchberger's algorithm over the dual noetherian valuation domain. We present some criterions that can be applied in the algorithm for computing Gröbner bases, and the criterions may drastically reduce the number of S-polynomials in the course of the algorithm. In addition, we clearly demonstrate the improvement with an example.</p></abstract>

Generalized lattices over one-dimensional noetherian domains

We study direct sum decompositions of pure projective torsion free modules over one-dimensional commutative noetherian domains. Drawing inspiration from the representation theory of orders in separable algebras, we study when every pure projective torsion free module is a direct sum of finitely generated modules. A satisfactory criterion is given for analytically unramified reduced local rings and for Bass domains.

Using Semicontinuity for Standard Bases Computations

We present new results on standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring over a computable field K. We prove the semicontinuity of the “highest corner” in a family of ideals, parametrized by the spectrum of a Noetherian domain A. This semicontinuity is used to design a new modular algorithm for computing a standard basis of I if K is the quotient field of A. It uses the computation over the residue field of a “good” prime ideal of A to truncate high order terms in the subsequent computation over K. We prove that almost all prime ideals are good, so a random choice is very likely to be good, and whether it is good is detected a posteriori by the algorithm. The algorithm yields a significant speed advantage over the non-modular version and works for arbitrary Noetherian domains. The most important special cases are perhaps A={mathbb {Z}} and A=k[t], k any field and t a set of parameters. Besides its generality, the method differs substantially from previously known modular algorithms for A={mathbb {Z}}, since it does not manipulate the coefficients. It is also usually faster and can be combined with other modular methods for computations in local rings. The algorithm is implemented in the computer algebra system Singular and we present several examples illustrating its power.

Restricted Minimum Condition in Reduced Commutative Rings

We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, the factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM domains generalize well. However, as we will show, RM rings need not be Noetherian and may have nilpotent elements. One of the classic results in the theory of RM rings is that for Noetherian domains the RM condition corresponds to having Krull dimension at most one. We will show that this can be generalized to reduced Noetherian rings, thus proving that affine rings corresponding to curves are RM. We will give examples showing that the assumption that the ring is reduced is not superfluous. In the second part, we will study CDR domains, i.e., domains where for any two ideals I, J the inclusion \(I\subseteq J\) implies that I is a multiple of J. We will prove that CDR domains are RM and this will allow us to give a new characterization of Dedekind domains. Examples of RM rings for various classes of rings will be given. In particular, we will show that a ring of polynomials R[x] is RM if and only if R is a reduced Artinian ring. And we will study the relation between RM rings and UFDs.