Related papers: Generalized Maximal Orders

We study the existence of maximal ideals in preadditive categories defining an order $\preceq$ between objects, in such a way that if there do not exist maximal objects with respect to $\preceq$, then there is no maximal ideal in the…

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two…

Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is…

In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We…

We give a full classification, up to equivalence, of finite-dimensional graded division algebras over the field of real numbers. The grading group is any abelian group.

While computer programs and logical theories begin by declaring the concepts of interest, be it as data types or as predicates, network computation does not allow such global declarations, and requires *concept mining* and *concept…

For a Dedekind domain $R$ with field of fractions $K$ a classical $R$-order in a semisimple $K$-algebra $A$ is an $R$-projective $R$-subalgebra $\Lambda$ of $A$ such that $K\Lambda=A$. We study differential graded $K$-algebras which are…

We prove the existence of an effective universal upper bound for the order of any integral periodic orbit of any integral algebraic dynamical system in a fixed ambient space. Using this, we demonstrate the decidability of periodicity in…

The question of embedding fields into central simple algebras $B$ over a number field $K$ was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields $L$ of such an algebra…

We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…

Let $K$ be a field, $\mathcal {O}_v$ a valuation ring of $K$ associated to a valuation $v$: $K\rightarrow\Gamma\cup\{\infty\}$, and ${\bf m}_v$ the unique maximal ideal of $\mathcal {O}_v$. Consider an ideal $\mathcal {I}$ of the free…

We describe the upper seminormal crystal structure for the $\mu$-supported $\delta$-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster $\mc{X}$-variety.…

Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set $P=\{p_1,...,p_n\}\subset\mathbb{Z}^2$ a maximal integral point set over $\mathbb{Z}^2$ if all pairwise distances are integral and…

Let $k$ be an arbitrary field. We classify the maximal reductive subgroups of maximal rank in any classical simple algebraic $k$-group in terms of combinatorial data associated to their indices. This result complements [S, 2022], which does…

The Dedekind tessellation -- the regular tessellation of the upper half-plane by the Mobius action of the modular group -- is usually viewed as a system of ideal triangles. We change the focus from triangles to circles and give their…

Let R be a finite principal left ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module R^n is produced. This is used to set up a greedy algorithm that selects vectors for which all…

In this paper we study orders over Cohen-Macaulay rings. We discuss desirable properties for these orders if they are to represent NCCRs of the base rings. While some definitions have been made, we discuss an alternate definition and the…

We consider dominant dimension of an order over a Cohen-Macaulay ring in the category of centrally Cohen-Macaulay modules. There is a canonical tilting module in the case of positive dominant dimension and we give an upper bound on the…

Let $T$ be a totally ordered set and let $D(T)$ denote the set of all cuts of $T$. We prove the existence of a discrete valuation domain $O_{v}$ such that $T$ is order isomorphic to two special subsets of Spec$(O_{v})$. We prove that if $A$…

The global dimension of a ring governs many useful abilities. For example, it is semi-simple if the global dimension is 0, hereditary if it is 1 and so on. We will calculate the global dimension of a Crystalline Graded Ring, as defined in…