Products of projective S-Systems

The tree main results of this paper deal with the generating vectors of simple infinite and finite dimesional modules arising as composition factors of the Kac modules K↓ G ₀ and K of the Lie superalgebra sl(m/n).     

Explicit linear minimal free resolutions over a natural class of Rees algebras

Homological properties of the Rees algebra R of a Koszul K-algebra A over a field K, with respect to the maximal homogeneous ideal, are studied. In particular, for a finitely generated graded A-module N with linear minimal free R-resolution over A, the minimal free resolution of is explicitly constructed. This resolution is again linear.Received: 23 October 2000     

Gaussian elements of a semicontent algebra

The connection between a univariate polynomial having locally principal content and the content function acting like a homomorphism (the so-called Gaussian property) has been explored by many authors. In this work, we extend several such results to the contexts of multivariate polynomials, power series over a Noetherian ring, and base change of affine K-algebras by separable algebraically closed field extensions. We do so by using the framework of the Ohm–Rush content function. The correspondence is particularly strong in cases where the base ring is approximately Gorenstein or the element of the target ring is regular.     

Cohen–Macaulay,Gorenstein, complete intersection and regular defect for the tensor product of algebras

The main goal of this paper is to measure the defect of Cohen–Macaulay, Gorenstein, complete intersection and regularity for the tensor product of algebras over a ring. For this sake, we determine the homological invariants which are inherent to these notions, such as the Krull dimension, depth, injective dimension, type and embedding dimension of the tensor product constructions in terms of those of their components. Our results allow to generalize various theorems in this topic especially [4, Theorem 2.1], [21, Theorem 6] and [14, Theorems 1 and 2] as well as two Grothendieck's theorems on the transfer of Cohen–Macaulayness and regularity to tensor products over a field issued from finite field extensions. To prove our theorems on the defect of complete intersection and regularity, the homology theory introduced by André and Quillen for commutative rings turns out to be an adequate and efficient tool in this respect.     

Topological properties of localizations,flat overrings and sublocalizations

We study the set of localizations of an integral domain from a topological point of view, showing that it is always a spectral space and characterizing when it is a proconstructible subspace of the space of all overrings. We then study the same problems in the case of quotient rings, flat overrings and sublocalizations.     

Trivial extensions defined by Prüfer conditions

This paper deals with well-known extensions of the Prüfer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zero-divisors subject to various Prüfer conditions. The new examples give further evidence for the validity of the Bazzoni-Glaz conjecture on the weak global dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.     

Componentwise regularity (I)

We define the notion of componentwise regularity and study some of its basic properties. We prove an analogue, when working with weight orders, of Buchberger's criterion to compute Gröbner bases; the proof of our criterion relies on a strengthening of a lifting lemma of Buchsbaum and Eisenbud. This criterion helps us to show a stronger version of Green's crystallization theorem in a quite general setting, according to the componentwise regularity of the initial object. Finally we show a necessary condition, given a submodule M   of a free one over the polynomial ring and a weight such that in(M)$\mathrm{in}(M)$ is componentwise linear, for the existence of an i   such that _βi(M)=_βi(in(M))${\beta }_i(M)={\beta }_i(\mathrm{in}(M))$.     

Gröbner bases of ideals cogenerated by Pfaffians

We characterise the class of one-cogenerated Pfaffian ideals whose natural generators form a Gröbner basis with respect to any anti-diagonal term order. We describe their initial ideals as well as the associated simplicial complexes, which turn out to be shellable and thus Cohen-Macaulay. We also provide a formula for computing their multiplicity.     

On the class group and S-class group of formal power series rings

Let ${\text{Cl}}_t(A)$ denote the t-class group of an integral domain A. P. Samuel has established that if A is a Krull domain then the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is injective and if A is a regular UFD, then ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, is bijective. Later, L. Claborn extended this result in case A is a regular Noetherian domain. In the first part of this paper we prove that the mapping ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$; $[I]?[{(I.A?X?)}_t]$ is an injective homomorphism and in case of an integral domain A such that each υ-invertible υ-ideal of A has υ-finite type, we give an equivalent condition for ${\text{Cl}}_t(A)\to {\text{Cl}}_t(A?X?)$, to be bijective, thus generalizing the result of Claborn. In the second part of this paper, we define the S-class group of an integral domain A: let S be a (not necessarily saturated) multiplicative subset of an integral domain A. Following [11], a nonzero fractional ideal I of A is S-principal if there exist an $s\in S$ and $a\in I$ such that $sI?aA?I$. The S-class group of A, S-${\text{Cl}}_t(A)$ is the group of fractional t-invertible t-ideals of A under t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of A. We generalize some known results developed for the classic contexts of Krull and PυMD domain and we investigate the case of isomorphism S-${\text{Cl}}_t(A)?S$-${\text{Cl}}_t(A?X?)$.     

On inverse-direct systems of modules

Inverse-direct systems of modules have been considered by Eklof and Mekler, see [P.C. Eklof, A.H. Mekler, Almost free modules, 2nd ed., North Holland, 2002]. The systems we are going to study are different: we do not assume the condition that certain composite maps are identity maps (this forces the direct summand property). In this paper inverse-direct systems will be considered where certain composite maps lie in the center of the respective endomorphism rings. We investigate how the limits are modified if the connecting maps are changed by automorphisms of the modules. It will also be shown that one can define a composition between the systems modified by these automorphisms such that those whose limits are non-isomorphic under the canonical maps form an abelian group. This group can be described in terms of the first derived functor of the inverse limit functor.We also study the relation to vanishing inverse limits: in certain cases, the maps can be modified in such a way that the inverse limit of the new system becomes 0. In the final section, we use self-idealizations in order to construct sets of non-isomorphic modules (over suitable uncountable rings) that are direct limits of the same collection of modules with different connecting maps.     

Note on non-D-rings

Abstract

Recall that an integral domain R is said to be a non-D-ring if there exists a non-constant polynomial f (X) in R[X] (called a uv-polynomial) such that f (a) is a unit of R for every a in R. In this note we generalize this notion to commutative rings (that are not necessarily integral domains) as follows: for a positive integer n, we say that R is an n-non-D-ring if there exists a polynomial f of degree n in R[X] such that f (a) is a unit of R for every a in R. We then investigate the properties of this notion in di?erent contexts of commutative rings.     

Cancellation properties in ideal systems: A classification of e.a.b. semistar operations

We give a classification of e.a.b. semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to partition the collection of all e.a.b. semistar (or star) operations, we show that there is exactly one operation of finite type in each equivalence class and that this operation has a range of nice properties. We give examples to demonstrate that the four classes of e.a.b. semistar (or star) operations we defined can all be distinct. In particular, we solve the open problem of showing that a.b. is really a stronger condition than e.a.b.     

A monomial cycle basis on Koszul homology modules

We give a class of p-Borel principal ideals of a polynomial algebra over a field K for which the graded Betti numbers do not depend on the characteristic of K and the Koszul homology modules have a monomial cyclic basis.     

Quasi-socle ideals and Goto numbers of parameters

Goto numbers for certain parameter ideals Q in a Noetherian local ring (A,m) with Gorenstein associated graded ring are explored. As an application, the structure of quasi-socle ideals I=Q:m^q (q≥1) in a one-dimensional local complete intersection and the question of when the graded rings are Cohen-Macaulay are studied in the case where the ideals I are integral over Q.     

Strongly irreducible ideals of a commutative ring

An ideal I of a ring R is said to be strongly irreducible if for ideals J and K of R, the inclusion J∩K⊆I implies that either J⊆I or K⊆I. The relationship among the families of irreducible ideals, strongly irreducible ideals, and prime ideals of a commutative ring R is considered, and a characterization is given of the Noetherian rings which contain a non-prime strongly irreducible ideal.     

Counter-examples to non-noetherian Elkik's approximation theorem

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies.