Using quantum sections of filtered rings and the associated Rees rings one can lift the scheme structure on Proj of the associated graded ring to the Proj of the Rees ring. The algebras of interest here are positively filtered rings having a non-commutative regular quadratic algebra for the associated graded ring; these are the so-called gauge algebras obtaining their name from special examples appearing in E. Witten's gauge theories. The paper surveys basic definitions and properties but concentrates...

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...

Let Φ be a system of ideals on a commutative Noetherian ring R, and let S be a multiplicatively closed subset of R. The first result shows that the topologies defined by ${I_a}_{I\in \Phi }$ and ${S\left(I_a\right)}_{I\in \Phi }$ are equivalent if and only if S is disjoint from the quintasymptotic primes of Φ. Also, by using the generalized Lichtenbaum-Hartshorne vanishing theorem we show that, if (R,) is a d-dimensional local quasi-unmixed ring, then $H_{\Phi }^d\left(R\right)$, the dth local cohomology module of R with respect to Φ, vanishes if and only if there exists...

In this article we formalize a quotient module of Z-module and a vector space constructed by the quotient module. We formally prove that for a Z-module V and a prime number p, a quotient module V/pV has the structure of a vector space over Fp. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattices [14]. Some theorems in this article are described by translating theorems in [20] and [19] into theorems of Z-module....