Related papers: Geometrically closed rings
In this paper we show how the hyperstructure concept leads to new algebraic structures and general field theories.
By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…
There is an interpretation of open string field theory in algebraic topology. An interpretation of closed string field theory can be deduced from this open string theory to obtain as well the interpretation of open and closed string field…
In this paper, we introduce an algebro-geometric formulation for Siegel's theorem using an improvement of Lang's version of Roth's theorem over finitely generated fields of characteristic zero. In fact, we prove that, for an affine open…
The book is devoted to investigation of arithmetic of the matrix rings over certain classes of commutative finitely generated principal ideals domains. We mainly concentrate on constructing of the matrix factorization theory. We reveal a…
We show that a class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular…
We prove that the main examples in the theory of algebraic differential equations possess a remarkable total differential overconvergence property. This allows one to consider solutions to these equations with coordinates in algebraically…
The focus of this article is on metric completions of triangulated categories arising in the representation theory of hereditary finite dimensional algebras and commutative rings. We explicitly describe all completions of bounded derived…
In classical geometric algebra, there have been several treatments of affine and projective planes based on fields. In this thesis we approach affine and projective planes from a constructive point of view and we base our geometry on local…
Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that…
The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is…
I extend the framework of rigid analytic geometry to the setting of algebraic geometry relative to monoids, and study the associated notions of separated, proper, and overconvergent morphisms. The category of affine manifolds embeds as a…
We show that the defining relations needed to describe a generalized q-Schur algebra as a quotient of a quantized enveloping algebra are determined completely by the defining ideal of a certain finite affine variety, the points of which…
In this paper, we propose an abstract definition of dependent type theories as essentially algebraic theories. One of the main advantages of this definition is its composability: simple theories can be combined into more complex ones, and…
It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let $\mathsf{G}$ be a group,…
Some basic properties of the ring of integers $\mathbb{Z}$ are extended to entire rings. In particular, arithmetic in entire principal rings is very similar than arithmetic in the ring of integers $\mathbb{Z}$. These arithmetic properties…
We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…
Generalized geometry finds many applications in the mathematical description of some aspects of string theory. In a nutshell, it explores various structures on a generalized tangent bundle associated to a given manifold. In particular,…
We explore connected affine algebraic groups $G$, which enjoy the following finiteness property $\rm (F)$: for every algebraic action of $G$, the closure of every $G$-orbit contains only finitely many $G$-orbits. We obtain two main results.…
Outside of the framework of geometric theories, we exhibit complete, respectively model-complete theories of rings whose corresponding theory of pairs is complete, respectively model-complete, using transfer results proven in the seventies…