Let $E$ be the ring of integer valued polynomials over $\mathbb{Z}$. This ring is known to be a Prüfer domain. But it seems there does not exist an algorithm for inverting a nonzero finitely generated ideal of $E$. In this note we show how to obtain such an algorithm by deciphering a classical abstract proof that uses localisations of $E$ at all prime ideals of $E$. This confirms a general program of deciphering abstract classical proofs in order to obtain algorithmic proofs.

Soit $R=k[x,y,z{]}_{(x,y,z)}$ le localisé de l’anneau des polynômes à trois variables sur le corps $k$ de caractéristique nulle. Nous construisons une valuation divisorielle $\nu$ de $R$, nous calculons un système minimal de générateurs de la $k$-algèbre ${\mathrm{gr}}_{\nu }\left(R\right)$ associée à la filtration $\nu$-adique. Ce système est infini : ${\mathrm{gr}}_{\nu }\left(R\right)$ n’est pas noethérien.

Un anneau gradué unitaire où tout élément homogène non nul est inversible est appelé un anneau à division gradué. Cet article est une contribution à l’étude de la correspondance existante entre les anneaux à division valués et les anneaux à division gradués , voir [1], [2], [3], [4], [6] et [7].Il a été prouvé dans [5, Remarque de la page 26], que toute extension gr-séparable finie de corps gradués n’est pas simple. Dans ce travail on donne un critère pour l’existence d’élément primitif dans une...

In un UFD ogni elemento non unitario $\ne 0$ può essere espresso in modo unico nella forma $u{p}_1^{a_1}\mathrm{\dots }{p}_n^{a_n}$ dove $u$ è un elemento unitario, i $p_i$ sono primi non associati e ogni $a_i\ge 1$. Per studiare questa fattorizzazione in un ambito non atomico, si prende in esame un certo numero di generalizzazioni della potenza di un primo $p^n$. Per numerose di queste generalizzazioni si prova che si ottiene una forma di fattorizzazione unica e la si mette in relazione, nel caso in cui $R$ è un dominio di integrità, con rappresentazioni di carattere...

The goal of this paper is to develop tools to study maximal families of Gorenstein quotients A of a polynomial ring R. We prove a very general theorem on deformations of the homogeneous coordinate ring of a scheme Proj(A) which is defined as the degeneracy locus of a regular section of the dual of some sheaf M of rank r supported on say an arithmetically Cohen-Macaulay subscheme Proj(B) of Proj(R). Under certain conditions (notably; M maximally Cohen-Macaulay and ∧r M ≈ KB(t) a twist of the canonical...

Let $R$ be an integral domain with quotient field $K$ and $f\left(x\right)$ a polynomial of positive degree in $K\left[x\right]$. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form $I=f\left(x\right)K\left[x\right]\cap R\left[x\right]$ are almost principal in the following two cases: – $J$, the ideal generated by the leading coefficients of $I$, satisfies $J^{-1}=R$. – $I^{-1}$ as the $R\left[x\right]$-submodule of $K\left(x\right)$ is of finite type. Furthermore we prove that for $I=f\left(x\right)K\left[x\right]\cap R\left[x\right]$ we have: – $I^{-1}\cap K\left[x\right]=\left(I{:}_{K\left(x\right)}I\right)$. – If there exists $p/q\in I^{-1}-K\left[x\right]$, then $(q,f)\ne 1$...