Related papers: On the Equational Artinian Algebras
In this article, the properties of being equational noetherian, $q_{\omega}$ and $u_{\omega}$-compactness, and equational Artinian are studied from the perspective of the Zariski topology. The equational conditions on the relative free…
Let $R$ be a unital commutative ring with unit and $\mathscr{G}$ an ample groupoid. Using the topology of the groupoid $\mathscr{G}$, Steinberg defined an etale groupoid algebra $R\mathscr{G}$. These etale groupoid algebras generalize…
In this article, we describe the relation between the properties of being equational noetherian and ascending chain condition on ideals of an arbitrary algebra. We also give a formulation of Hilbert's basis theorem for varieties of algebras…
In this article, we study the property of being equationally Artinian in groups. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian…
A duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra is established. Let $A$ be a noetherian complete basic semiperfect algebra over an algebraically closed field, and $C$ be its dual…
Octonion algebras are certain algebras with a multiplicative quadratic form. In their 2019 article, Alsaody and Gille show that, for octonion algebras over unital commutative rings, there is an equivalence between isotopes and isometric…
This article concerns commutative algebras over a field $k$ of characteristic zero which are finite dimensional as vectorspaces, and particularly those of such algebras which are graded. Here the term graded is applied to non-negatively…
We consider algebras over a field K defined by a presentation K <x_1,..., x_n : R >, where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one of…
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…
This paper proposes a new category theoretic account of equationally axiomatizable classes of algebras. Our approach is well-suited for the treatment of algebras equipped with additional computationally relevant structure, such as ordered…
Given an Artinian algebra $A$ over a field $k$, there are several combinatorial objects associated to $A$. They are the diagram $D_A$ as defined in [DK], the natural quiver $\Delta_A$ defined in \cite{Li} (cf. Section 2), and a generalized…
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 define the Artinian and Noetherian algebra which consist of formal series involving exponents which are not necessarily integers. All of the usual operations are defined here and characterized. As an application, we compute the algebra…
In this work, we introduce a new class of Leibniz algebras, called quasi-Artinian Leibniz algebras, which generalizes the minimal condition on ideals. Furthermore, we provide some characterizations and give conditions under which a…
Universal algebraic geometry is generalised from solutions of equations in a single algebra to the study of $\varphi$- or $K$-spectra, akin to the prime spectrum of a ring. We explore their basic properties and constructions, give a…
The motivation of this paper is to study the natural quiver of an artinian algebra, a new kind of quivers, as a tool independing upon the associated basic algebra. In \cite{Li}, the notion of the natural quiver of an artinian algebra was…
Let $A$ be a partial *-algebra endowed with a topology $\tau$ that makes it into a locally convex topological vector space $A[\tau]$. Then $A$ is called a topological partial *-algebra if it satisfies a number of conditions, which all…
Following a previous work with Boudi, we continue to investigate Bernstein algebras satisfying chain conditions. First, it is shown that a Bernstein algebra $(A, \omega)$ with ascending or descending chain condition on subalgebras is…
The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…
The theory of path algebras is usually circunscripted to the study of representations, usually linked to finite graphs. In our work, we focus on studying the structure of path algebras over a field associated to arbitrary graphs. We…