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.

     

Prime ideals of principally quasi-Baer rings

We extend a theorem of Kist for commutative PP rings to principally quasi-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. Also decompositions of quasi-Baer and principally quasi-Baer rings are investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Prime ideals in restricted differential operator rings

In this paper restricted differential operator rings are studied. A restricted differential operator ring is an extension of ak-algebraR by the restricted enveloping algebra of a restricted Lie algebra g which acts onR. This is an example of a smash productR #H whereH=u (g). We actually deal with a more general twisted construction denoted byR * g where the restricted Lie algebra g is not necessarily embedded isomorphically inR * g. Assume that g is finite dimensional abelian. The principal result obtained is Incomparability, which states that prime idealsP ₁ ⊆P ₂ ⊂R * g have different intersections withR. We also study minimal prime ideals ofR * g whenR is g-prime, showing that the minimal primes are precisely those having trivial intersection withR, that these primes are finite in number, and their intersection is a nilpotent ideal. Prime and primitive ranks are considered as an application of the foregoing results.     

On prime ideals and radicals of polynomial rings and graded rings

We extend some known results on radicals and prime ideals from polynomial rings and Laurent polynomial rings to Z$\mathbb{Z}$-graded rings, i.e, rings graded by the additive group of integers. The main of them concerns the Brown–McCoy radical G$G$ and the radical S$S$, which for a given ring A$A$ is defined as the intersection of prime ideals I$I$ of A$A$ such that A/I$A/I$ is a ring with a large center. The studies are related to some open problems on the radicals G$G$ and S$S$ of polynomial rings and situated in the context of Koethe’s problem.     

Prime ideals in polynomial rings in several indeterminates

If is a prime ideal of a polynomial ring , where is a field, then is determined by an irreducible polynomial in . The purpose of this paper is to show that any prime ideal of a polynomial ring in -indeterminates over a not necessarily commutative ring is determined by its intersection with plus polynomials.

     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

Group graded associated ideals with flat base change of rings and short exact sequences

This paper deals with the study of behaviour of G-associated ideals and strong Krull G-associated ideals with flat base change of rings and behaviour of G-associated ideals with short exact sequences over rings graded by finitely generated abelian group G.     

Prime graded modules

In this paper, properties of prime and strongly prime graded modules are studied. The class of strongly prime graded modules that determines a graded strongly prime radical is described.