Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate quadratic...

The notion of a closed polynomial over a field of zero characteristic was introduced by Nowicki and Nagata. In this paper we discuss possible ways to define an analog of this notion over fields of positive characteristic. We are mostly interested in conditions of maximality of the algebra generated by a polynomial in a respective family of rings. We also present a modification of the condition of integral closure and discuss a condition involving partial derivatives.

Let 𝓐(ℝ) and 𝓔(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if 𝔪 is a maximal ideal of 𝓐(ℝ), then 𝓐(ℝ)/𝔪 is isomorphic either to the reals or a real closed field that is an η₁-set, while if 𝔪 is a maximal ideal of 𝓔(ℝ), then 𝓔(ℝ)/𝔪 is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of...

A ring extension $R\subseteq S$ is said to be FO if it has only finitely many intermediate rings. $R\subseteq S$ is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension $R\subseteq S$ to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair...

Soient $k$ un corps commutatif et $I=(p_1,\cdots ,p_m)k_n\left[X\right]$ un idéal de l’anneau des polynômes $k[X_1,\cdots ,X_n]$ (éventuellement $I=k_n\left[X\right]$). Nous prouvons une conjecture de C. Berenstein - A. Yger qui affirme que pour tout polynôme $p$, élément de la clôture intégrale $\overline{I}$ de l’idéal $I$, on a une représentation$$p^m=\sum _{1\le i\le m}{p}_i{q}_i,\text{avec}maxdeg\left(q_i{p}_i\right)\le mdegp+m{d}_1\cdots d_m,$$où $d_i=deg{p}_i,1\le i\le m$.

Let $I$ and $J$ be ideals of a Noetherian local ring $(R,𝔪)$ and let $M$ be a nonzero finitely generated $R$-module. We study the relation between the vanishing of $H_{I,J}^{dimM}\left(M\right)$ and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian $R$-module $M/JM$ is equal to its integral closure relative to the Artinian $R$-module $H_{I,J}^{dimM}\left(M\right)$.

Using the flatification by blow-up result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination...

Sia a un intero algebrico con il polinomio minimale $f\left(X\right)$. Si danno condizioni necessarie e sufficienti affinché l'anello $\mathbb{Z}\left[\alpha \right]$ sia seminormale o $t$-chiuso per mezzo di $f\left(X\right)$. Come applicazione, in particolare, si ottiene che se $f\left(X\right)=X^3+aX+b$, $a$, $b\in \mathbb{Z}$ le condizioni sono espresse mediante il discriminante de $f\left(X\right)$.