We study relative algebraic K-theory of admissible Zariski-Riemann spaces and prove that it is equivalent to G-theory satisfies homotopy invariance. Moreover, we provide an example a non-noetherian abelian category whose negative vanishes.

We introduce the notion of a Hu-Liu prime ideal in the context of left commutative rngs, and establish the contravariant functor from the category of left commutative rngs into the category of topological spaces. It is well known that new points must be introduced in order to expand algebraic geometry over algebraically closed fields into Grothendieck’s scheme theory over commutative rings. We ...

Dans la section 4f concernant plusieurs plonge-l distincts de Z l est encore distincte de Z l : si B est une variitt ablienne sur un corps K, et si est un sous-groupe de B(K) Zariski dense dans B, il existe 2 qui engendre un sous-groupe Z Zariski dense dans B.. Comme me l'a fait remarquer J-P. Serre, Z 2 est rrunion de trois sous-groupes d'indice 2, tels que (2Z)Z, Z(2Z) et Z(1; 1)+Z(1; 1). D'a...

The prime k-spectrum Speck(R) of a k-semiring R will be introduced. It will be proven that it is a topological space, and some properties of this space will be investigated. Connections between the topological properties of Speck(R) and possible algebraic properties of the k-semiring R will be established.

Let $R$ be a commutative ring and let $M$ be an $R$-module. In this article, we introduce the concept of the Zariski socles of submodules of $M$ and investigate their properties. Also we study modules with Noetherian second spectrum and obtain some related results.

in this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.