Related papers: Hidden constructions in abstract algebra, Krull Di…
We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espa\~nol and the authors. We show that the notion of Krull…
Nous rappelons des versions constructives de la th\'eorie de la dimension de Krull dans les anneaux commutatifs et dans les treillis distributifs, dont les bases ont \'et\'e pos\'ees par Joyal, Espan\~ol et les deux auteurs. Nous montrons…
This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules, which constitutes the algebraic version of the vector bundles in differential geometry. We adopt the…
This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules. We adopt the constructive point of view, with which all existence theorems have an explicit algorithmic…
The aim of this survey is to present applications of covering techniques in the theory of Krull-Gabriel dimension. We start with recalling fundamental facts of the classical covering theory of quivers and locally bounded categories. Then we…
We develop some algebraic structure notions such as composition series and convexity degree, along with some notions holding a geometric interpretation, like reducibility and hyperdimension, with the main objective being a tropical…
The first part of the present article consists in a survey about the dynamical constructive method designed using dynamical theories and dynamical algebraic structures. Dynamical methods uncovers a hidden computational content for numerous…
There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and…
In this paper, we extend properties Going Up and Lying Over from ring theory to the general setting of congruence--modular equational classes, using the notion of prime congruence defined through the commutator. We show how these two…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
This work demonstrates classical generation is preserved by the derived pushforward along the structure morphism of a noncommutative coherent algebra to its underlying scheme. Additionally, we establish that the Krull dimension of a variety…
We determine the Krull-Gabriel dimension of weighted surface algebras, a class of algebras which recently appeared in the context of classification of tame symmetric periodic algebras of non-polynomial growth. Moreover, we consider…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
We construct functorially a class of algebras using the formalism of double derivations. These algebras extend to higher dimensions Crawley-Boevey and Holland's construction of deformed preprojective algebras and encompass symplectic…
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial…
AF-rings are algebras over a field k which satisfy the Altitude Formula over k. This paper surveys a few works in the literature on the Krull and valuative dimensions of tensor products of AF-rings. The first section extends Wadsworth's…
Let $A$ be a commutative arithmetical ring. The ring $A$ has Krull dimension if and only if every factor ring of $A$ is finite-dimensional and does not have idempotent proper essential ideals. The study is supported by Russian Science…
This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a closed model category structure. This model structure is---when working over an algebraically closed field of characteristic…
We prove constructively a Nullstellensatz giving an equivalence between the existence of a certain kind of algebraic identity on one hand, and the impossibility of finding an increasing sequence of irreducible varieties obeying certain…
The aim of this paper to draw attention to several aspects of the algebraic dependence in algebras. The article starts with discussions of the algebraic dependence problem in commutative algebras. Then the Burchnall-Chaundy construction for…