Related papers: Indecomposable Algebraic Integers
The aim of this paper is to investigate further properties of $z$-elements in multiplicative lattices. We utilize $z$-closure operators to extend several properties of $z$-ideals to $z$-elements and introduce various distinguished…
In this paper, a question due to Heckenberger, Shareshian and Welker on racks in [7] is positively answered. A rack is a set together with a selfdistributive bijective binary operation. We show that the lattice of subracks of every finite…
We prove the following result: Theorem. Every algebraic distributive lattice D with at most $\aleph\_1$ compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the $\aleph\_1$…
We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper…
Integer geometry on a plane deals with objects whose vertices are points in $\mathbb Z^2$. The congruence relation is provided by all affine transformations preserving the lattice $\mathbb Z^2$. In this paper we study circumscribed circles…
A lattice L is called opc if every monotone function f : L^n -> L is induced by a polynomial. We show here: If L is a lattice with the interpolation property whose cardinality is a strong limit cardinal of uncountable cofinality, then some…
We study how the existence in an algebraic lattice $L$ of a chain of a given type is reflected in the join-semilattice $K(L)$ of its compact elements. We show that for every chain $\alpha$ of size $\kappa$, there is a set $\B$ of at most…
We study the structure of the codifferent and of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal trace of indecomposable integers multiplied by totally…
We provide a sufficient condition for a polynomial ring, not necessarily commutative, to have a first-order definition for the rational integers.
Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated…
It is known from Grzegorczyk's paper \cite{grze-1951} that the lattice of real semi-algebraic closed subsets of ${\mathbb R}^n$ is undecidable for every integer $n\geq 2$. More generally, if $X$ is any definable set over a real or…
Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent…
In this paper we investigate the factorization behaviour of the binomial polynomials $\binom{x}{n} = \frac{x(x-1)\cdots (x-n+1)}{n!}$ and their powers in the ring of integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. While it is…
Let $R$ be a commutative integral unital domain and $L$ a free non-commutative Lie algebra over $R$. In this paper we show that the ring $R$ and its action on $L$ are 0-interpretable in $L$, viewed as a ring with the standard ring language…
In a previous paper by the author a universal ring of invariants for algebraic structures of a given type was constructed. This ring is a polynomial algebra that is generated by certain trace diagrams. It was shown that this ring admits the…
We study some interesting properties of Furstenberg's topology of the integers. We show that it is metrizable, totally disconnected, and (Z,+,.) is a topological ring with respect to this topology. As an application, we show that any two…
We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…
Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. In this paper we introduce and study similar notions for algebras. Let $k$ be a field and let $A$ be an…
In the paper we study the algebroid A of the groupoid of partially invertible elements over the lattice of orthogonal projections of a $W^*$-algebra. In particular the complex analytic manifold structure of these objects is investigated.…
Algebraic integers in totally imaginary quartic number fields are not discrete in the complex plane under a fixed embedding, which makes it impossible to visualize all integers in the plane, unlike the quadratic imaginary algebraic…