Related papers: The Transcendence Degree over a Ring

Let $E$ be a module of projective dimension one over a Noetherian ring $R$ and consider its Rees algebra $\mathcal{R}(E)$. We study this ring as a quotient of the symmetric algebra $\mathcal{S}(E)$ and consider the ideal $\mathcal{A}$…

We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary…

Let $\Omega$ be a class of unital ${\rm C^*}$-algebras which have the second type tracial nuclear dimensional at moat $n$ (or have tracial nuclear dimensional at most $n$). Let $A$ be an infinite dimensional unital simple ${\rm…

We introduce a new class of quasi-hereditary algebras, containing in particular the Auslander-Dlab-Ringel (ADR) algebras. We show that this new class of algebras is preserved under Ringel duality, which determines in particular explicitly…

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place. We show an elementary algebraic approach to…

The main result of the paper is the classification of all (nonassociative) algebras of level two, i.e. such algebras that maximal chains of nontrivial degenerations starting at them have length two. During this classification we obtain an…

Let $k$ be a field of positive characteristic $p$, $R$ be a Gorenstein graded $k$-algebra, and $S=R/J$ be an artinian quotient of $R$ by a homogeneous ideal. We ask how the socle degrees of $S$ are related to the socle degrees of…

We establish a connection between two areas of independent interest in representation theory, namely Koszul duality and higher homological algebra. This is done through a generalization of the notion of $T$-Koszul algebras, for which we…

We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and…

Let $R$ be a positively graded algebra over a field. We say that $R$ is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical…

We study the structure of a $3-$Leibniz algebra $T$ graded by an arbitrary abelian group $G,$ which is considered of arbitrary dimension and over an arbitrary base field $\bbbf.$ We show that $T$ is of the form $T=\uu\oplus\sum_jI_j,$ with…

In this article we classify indecomposable objects of the derived categories of finitely-generated modules over certain infinite-dimensional algebras. The considered class of algebras (which we call nodal algebras) contains such well-known…

In the present paper, we give sufficient conditions on the elements of the continued fractions $A$ and $B$ that will assure us that the continued fraction $A^B$ is a transcendental number. With the same condition, we establish a…

In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a series of positive rational terms is a transcendental number.…

Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra, and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented…

Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring $A\otimes_k k(t)_{per}$ is noetherian or not. As a consequence, we prove that the ring…

Let $A$ be a coherent algebra and $B$ be a finite-dimensional Gorenstein algebra over a field $k$. We describe finitely presented Gorenstein projective $A\otimes_k B$-modules in terms of their underlying onesided modules. Moreover, if the…

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of…

The rate of a standard graded $K$-algebra $A$ is a measure of the growth of the shifts in a minimal free resolution of $K$ as an $A$-module. In particular $A$ has rate one if and only if it is Koszul. It is known that a generic Artinian…

The lower transcendence degree, introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable…