Related papers: Zero Divisors in Associative Algebras over Infinit…
Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified.
This paper addresses the classification problem of associative algebras over arbitrary base fields. We present a list of three-dimensional associative algebras with canonical representatives of the isomorphism classes for fields of…
To allow for Division By Zero, we develop a new algebraic structure containing addition and multiplication called an S-Extension of a Field. This unique structure extends a Field so that the equation $0\cdot s=x$ has exactly one solution…
Let $F$ be a field of characteristic not $2$ . An associative $F$-algebra $R$ gives rise to the commutator Lie algebra $R^{(-)}=(R,[a,b]=ab-ba).$ If the algebra $R$ is equipped with an involution $*:R\rightarrow R$ then the space of the…
The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a $k$-algebra and this new ``$k$-space'' becomes a generalization of the…
The genus gen(D) of a finite-dimensional central division algebra D over a field F is defined as the collection of classes [D'] in the Brauer group Br(F), where D' is a central division F-algebra having the same maximal subfields as D. For…
For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by a non-zero element are shown to form an invariant of A, called its double sign. The…
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…
We show that there are no symmetric non-zero biderivations on perfect Lie algebras of finite dimension over a field of characteristic zero. We show that this is equivalent to show that every symmetric biderivation on a finite-dimensional…
A celebrated theorem of P.M.Cohn says that for any two division rings (not necessarily finite dimensional) over a field F, their amalgamated product over F is a domain which can be embedded in a division ring. Note that even with the two…
Suppose that $X$ is a projective variety over an algebraically closed field of characteristic $p > 0$. Further suppose that $L$ is an ample (or more generally in some sense positive) divisor. We study a natural linear system in $|K_X + L|$.…
Fusion categories are fundamental objects in quantum algebra, but their definition is narrow in some respects. By definition a fusion category must be k-linear for some field k, and every simple object V is strongly simple, meaning that (V)…
Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. We say that $A$ is defined over a subfield $F_0$ of $F$ if $A = A_0\otimes_{F_0} F$ for some $F_0$-subalgebra $A_0$ of $A$. We show that: (1) In…
It was proved that whenever $\mathbb{N}$ is partitioned into finitely many cells, one cell must contain arbitrary length arithmetic and geometric progression nicely intertwined, so that one cell must be rich in the sense of containing…
The purpose of this paper is to study fields whose multiplicative groups admit the structure of linear spaces. We prove that the multiplicative group of a finite field is a linear space if and only if the order of the multiplicative group…
Modulo the ideal generated by the derivative fields, the normal ordered product of holomorphic fields in two-dimensional conformal field theory yields a commutative and associative algebra. The zero mode algebra can be regarded as a…
Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair of homogeneous zero divisors. As a consequence, we prove that…
We prove the result in the title. We infer, that unlike cylindric algebras, there is a first order axiomatization of the class of completely representable polyadic algebras of infinite dimension, though the one we obtain is infinite; in…
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…
The paper concerns perfect diassociative algebras and their implications to the theory of central extensions. It is first established that perfect diassociative algebras have strong ties with universal central extensions. Then, using a…