All rings considered in this paper are assumed to be commutative with identities. A ring $R$ is a $Q$-ring if every ideal of $R$ is a finite product of primary ideals. An almost $Q$-ring is a ring whose localization at every prime ideal is a $Q$-ring. In this paper, we first prove that the statements, $R$ is an almost $ZPI$-ring and $R\left[x\right]$ is an almost $Q$-ring are equivalent for any ring $R$. Then we prove that under the condition that every prime ideal of $R\left(x\right)$ is an extension of a prime ideal of $R$, the ring $R$ is a (an almost)...

Une somme amalgamée de schémas est décrite localement par un produit fibré d’anneaux. Ce texte donne un résultat global d’existence (§5.4) de schémas définis comme certaines sommes amalgamées et un procédé algébrique (§2.2) pour décrire les modules sur produits fibrés d’anneaux correspondants.

A ternary ring is an algebraic structure $ℛ=(R;t,0,1)$ of type $(3,0,0)$ satisfying the identities $t(0,x,y)=y=t(x,0,y)$ and $t(1,x,0)=x=(x,1,0)$ where, moreover, for any $a$, $b$, $c\in R$ there exists a unique $d\in R$ with $t(a,b,d)=c$. A congruence $\theta$ on $ℛ$ is called normal if $ℛ/\theta$ is a ternary ring again. We describe basic properties of the lattice of all normal congruences on $ℛ$ and establish connections between ideals (introduced earlier by the third author) and congruence kernels.

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$. Our focus is on homological properties of contracting endomorphisms of $R$, e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$-finite and $C$ is a semidualizing $R$-complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$-complex; (ii) $C\sim 𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$ for some $n>0$; (iii) ${\mathrm{G}}_C{\text{-dim}}^{n}R<\infty$ and $C$ is derived $𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$-reflexive...

Soit $G$ un groupe algébrique semi-simple simplement connexe défini sur un corps algébriquement clos $𝕜$ de caractéristique positive. Nous donnons une nouvelle preuve de l’existence d’un scindage de Frobenius de la variété des drapeaux de $G$ ainsi que de la nature $G$-équivariante de celui-ci. L’outil principal est un scindage de l’endomorphisme de Frobenius défini sur toute l’algèbre des distributions de $G$ qui permet de « détordre » la structure des $G$-modules.

Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual $E^{\text{'}}$ of a locally convex space $E$ is the $\sigma (E^{\text{'}},E)$-closure of the union of countably many $\sigma (E^{\text{'}},E)$-relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.