Principal Borel ideals and Gotzmann ideals

In this paper we characterize all principal Borel ideals with Borel generator up to degree 4 which are Gotzmann. We also classify principal Borel ideals with a Borel generator of degree d which are lexsegment and we describe the shadows of principal Borel ideals. Finally, we discuss the corresponding results for squarefree monomial ideals.Received: 10 May 2002     

On annihilator ideals of a polynomial ring over a noncommutative ring

Let R be a ring and let R[x] denote the polynomial ring over R. We study relations between the set of annihilators in R and the set of annihilators in R[x].     

Full ideals of polynomial rings

An idealI of the ringK[x ₁, ...,x _n ] of polynomials over a fieldK inn indeterminates is a full ideal ifI is closed under substitution,f I,g ₁...g_n K[x ₁, ...,x _n ] implyf(g ₁, ...,g _n ) I. In this paper we continue the investigation of full ideals ofK[x ₁, ...,x _n ]. In particular we determine several classes of full ideals ofK[x, y] (K a finite field) and investigate properties of these classes.The first author gratefully acknowledges support from theDeutsche Forschungsgemeinschaft     

Invariant ideals and polynomial forms

Let denote the group algebra of an infinite locally finite group . In recent years, the lattice of ideals of has been extensively studied under the assumption that is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra stable under the action of an appropriate automorphism group of . Specifically, in this paper, we let be a quasi-simple group of Lie type defined over an infinite locally finite field , and we let be a finite-dimensional vector space over a field of the same characteristic . If acts nontrivially on by way of the homomorphism , and if has no proper -stable subgroups, then we show that the augmentation ideal is the unique proper -stable ideal of when . The proof of this result requires, among other things, that we study characteristic division rings , certain multiplicative subgroups of , and the action of on the group algebra , where is the additive group . In particular, properties of the quasi-simple group come into play only in the final section of this paper.

     

Prime ideals in polynomial rings

Summary Study of relations between the prime and maximal spectra of a ringA and ofA[X], without noetherian assumptions. Application to the cases whereA has finite noetherian type andA is an arbitrary valuation domain; behaviour of the catenary property. New proofs of known results aboutG-ideals and Hilbert domains.

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Riassunto Si studiano le relazioni fra lo spettro ideale e quello massimale di un anelloA e diA[X] senza ipotesi di noetherianità. Si fanno delle applicazioni ai casi in cuiA è un anello di tipo noetheriano finito o è un arbitrario dominio di valutazione; si studia inoltre il comportamento della proprietà catenaria. Si danno nuove dimostrazioni di risultati noti suG-ideali e domini di Hilbert.

     

Explicit representation of membership in polynomial ideals

We introduce a new division formula on projective space which provides explicit solutions to various polynomial division problems with sharp degree estimates. We consider simple examples as the classical Macaulay theorem as well as a quite recent result by Hickel, related to the effective Nullstellensatz. We also obtain a related result that generalizes Max Noether’s classical AF + BG theorem.     

Radically perfect prime ideals in polynomial rings

We call an ideal I of a commutative ring R radically perfect if among the ideals of R whose radical is equal to the radical of I the one with the least number of generators has this number of generators equal to the height of I. Let R be a Noetherian integral domain of Krull dimension one containing a field of characteristic zero. Then each prime ideal of the polynomial ring R[X] is radically perfect if and only if R is a Dedekind domain with torsion ideal class group. We also show that over a finite dimensional Bézout domain R, the polynomial ring R[X] has the property that each prime ideal of it is radically perfect if and only if R is of dimension one and each prime ideal of R is the radical of a principal ideal.     

The lattice of ideals of a polynomial semiring

We show that for a semiringR, the following statements are equivalent: (1)R is a ring, (2) every left ideal ofR[X], the semiring of polynomials overR, is subtractive, (3) the lattice of left ideals ofR[X] is modular.Presented by R. W. Quackenbush.     

A negative answer to a question about leading terms ideals of polynomial ideals

We construct an example of a finitely generated ideal $I$ of $R\left[X\right]$, where $R$ is a one-dimensional domain, whose leading terms ideal is not finitely generated. This gives a negative answer to the open question of whether if $R$ is a domain with Krull dimension ≤1, then for any finitely generated ideal $I$ of $R\left[X\right]$, the leading terms ideal of $I$ is also finitely generated. Moreover, as a positive part of our answer, we prove that for any one-dimensional domain $R$ and any $a,b\in R$, the ideal of $R\left[X\right]$ generated by the leading terms of $〈1+aX,b〉$ is finitely generated.     

Computation of generalized real radicals of polynomial ideals

For an ordered field (K,T) and an idealI of the polynomial ring , the construction of the generalized real radical ofI is investigated. When (K,T) satisfies some computational requirements, a method of computing is presented. Project supported by the National Natural Science Foundation of China (Grant No. 19661002) and the Climbing Project.     

Left ideals of polynomial rings in several indeterminates

We consider maximal left ideals L of the polynomial ring R[λ₁, …, λ_n], for R noncommutative. In §1 we reprove and generalize Resco's result that any maximal left ideal L is generated by ≤ n elements whenever R is simple Artinian, and obtain more precise information about the generators when R satisfies a PI. In many instances, fewer than n generators suffice; this is considered in §3, by means of various examples. In §2 we see by a straightforward argument that L has bounded height as a prime left ideal whenever R is a simple Pl-ring, but this does not- hold in general for R simple Artinian.     

Coherent sequences of polynomial ideals on Banach spaces

This article deals with the relationship between an operator ideal and its natural polynomial extensions. We define the concept of coherent sequence of polynomial ideals and also the notion of compatibility between polynomial and operator ideals. We study the stability of these properties for maximal and minimal hulls, adjoint and composition ideals. We also relate these concepts with conditions on the underlying tensor norms (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)