Related papers: On the Equational Artinian Algebras
We obtain criteria for when a ring with enough idempotents is left/right artinian or noetherian in terms of local criteria defined by the associated complete set of idempotents for the ring. We apply these criteria to object unital category…
Algebraic theories, sometimes called equational theories, are syntactic notions given by finitary operations and equations, such as monoids, groups, and rings. There is a well-known category-theoretic treatment of them that algebraic…
The paper is essentially a continuation of B.Plotkin, G.Zhitomirski, "Some logical invariants of algebras and logical relations between algebras", St.Peterburg Math. J., {19:5}, (2008) 859 -- 879, whose main notion is that of…
We give a new and elementary proof of the nested Artin approximation Theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the…
Let $A$ be a nondegenerate dimer (or ghor) algebra on a torus, and let $Z$ be its center. Using cyclic contractions, we show the following are equivalent: $A$ is noetherian; $Z$ is noetherian; $A$ is a noncommutative crepant resolution;…
(1) Let 1\leq k\leq \omega. Call an atom structure \alpha weakly k neat representable, the term algebra is in \RCA_n\cap \Nr_n\CA_{n+k}, but the complex algebra is not representable. Call an atom structure neat if there is an atomic algebra…
In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important…
In this paper, we define on one hand, the notions of characteristics as well as central characteristics ideals of a given Leibniz algebra g and provide a necessary condition under which for two given subalgebras J and K of g such that, J IS…
First, we prove an algebraization result for rig-smooth algebras over a general noetherian ring; this positively answers the question raised in [Sta24, Tag 0GAX]. Then we prove a general partial algebraization result in non-archimedean…
Non notherian Formal schemes of perfectoid type (for example $\mathbb{Z}_p[p^{1/p^\infty}]\langle X^{1/p^\infty} \rangle$ along with its multivariate version) with rational degree are constructed and are shown to be admissible. These formal…
It is now a classical result that an algebraic space locally of finite type over $\mathbf{C}$ is analytifiable if and only if it is locally separated. In this paper we study non-archimedean analytifications of algebraic spaces. We construct…
This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The…
This article is about equationally Noetherian and weak equationally Noetherian property of Ershov algebras. Here we show two canonical forms of the system of equations over Ershov algebras and two criteria of equationally Noetherian and…
Using notions of homogeneity we give new proofs of M. Artin's algebraicity criteria for functors and groupoids. Our methods give a more general result, unifying Artin's two theorems and clarifying their differences.
Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected (but different from) group homology. It also gives a version of algebraic $K$-theory for rings by the simple functorial mapping assigning to…
Given two baric algebras $(A_1,\omega_1)$ and $(A_2,\omega_2)$ we describe a way to define a new baric algebra structure over the vector space $A_1\oplus A_2$, which we shall denote $(A_1\bowtie A_2,\omega_1\bowtie\omega_2)$. We present…
We will study the relationship of quite different object in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, $\tau$-categories and almost abelian categories. We will apply our results…
In this paper we discuss some special generalizations of equationally Noetherian property which naturally arise in the universal algebraic geometry. We introduce weakly equationally Noetherian, qw-compact, uw-compact, and weakly uw-compact…
To an artin algebra with radical square zero, a regular algebra in the sense of von Neumann and a family of invertible bimodules over the regular algebra are associated. These data describe completely, as a triangulated category, the…
Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic…