We study 0-dimensional real rank one valuations centered in a regular local ring of dimension n > 2 such that the associated valuation ring can be obtained from the regular ring by a sequence of quadratic transforms. We define two classical invariants associated to the valuation (the refined proximity matrix and the multiplicity sequence) and we show that are equivalent data of the valuation.

The concept of a Prüfer ring is studied in the case of rings with involution such that it coincides with the corresponding notion in the case of commutative rings.

Viene data una condizione sufficiente affinchè un sopra-anello di un anello di pseudo-valutazione (PVR) sia ancora un PVR. Da ciò segue che se $\left(R,M\right)$ è un PVR, allora ogni sopra-anello di $R$ è un PVR se (e soltanto se) $R\left[u\right]$ è quasi-locale per ciascun elemento $u$ di $\left(M:M\right)$. Vari risultati sono dimostrati per un ideale primo di un anello commutativo arbitrario $R$, avente $Z\left(R\right)$ come insieme di zero-divisori. Per esempio, se $P$ è un primo «forte» di $R$ e contiene un elemento non-zero divisore di $R$, allora $\left(P:P\right)$ è un sopra-anello...

In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if $g:Y\to X$ is a quasi-homeomorphism, $Z$ a sober space and $f:Y\to Z$ a continuous map, then there exists a unique continuous map $F:X\to Z$ such that $F\circ g=f$. Let $X$ be a $T_0$-space, $q:X\to {}^{s}X$ the injection of $X$ onto its sobrification ${}^{s}X$. It is shown, here, that $q\left(\text{Gold}\left(X\right)\right)=\text{Gold}\left({}^{s}X\right)$, where $\text{Gold}\left(X\right)$ is the set of all locally closed points of $X$. Some applications are also indicated. The Jacobson prime spectrum...

Soit $K$ un corps commutatif. Chercher une série formelle $S(X,T)\in K\left[\right[X,T\left]\right]$ vérifiant $S(X+Y,T)/S(X,T)\in K\left[\right[Y,T\left]\right]$ conduit naturellement à étudier l’application $U\left(T\right)\to \left(U\right(T){)}^X$, $U\left(T\right)$ étant une unité de l’algèbre $K\left[\right[T\left]\right]$, et à ramener les solutions à la forme $S(X,T)=\sum _{n\ge 0}{H}_n\left(X\right)T^n$, $\left(H_n\right(X\left)\right)$ étant une suite de $K\left[X\right]$ vérifiant les “identités multinomiales” :$$\left(\mu \right)H_n(X_1+...+X_k)=\sum _{{\alpha }_1+...+{\alpha }_k=n}{H}_{{\alpha }_1}(X_1)...H_{{\alpha }_k}(X_k\left)\right(n,k\ge 0).$$Après mise à l’écart par des lemmes combinatoires du cas caract$\left(K\right)\>0$ (les solutions sont triviales), on caractérise de plusieurs manières les solutions. On peut les faire coïncider avec l’ensemble NW des suites de polynômes (ou séries génératrices...