Related papers: Division in Associative $D$-Algebra
In this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra $Q$ is an algebra of…
We define a left quotient as well as a right quotient of two bounded operators between Hilbert spaces, and we parametrize these two concepts using the Moore-Penrose inverse. In particular, we show that the adjoint of a left quotient is a…
In the paper I considered algebra of polynomials over associative D-algebra with unit. Using the tensor notation allows to simplify the representation of polynomial. I considered questions related to divisibility of polynomial of any power…
Based on a class of associative algebras with zero-divisors which are called real-like algebras by us, we introduce a way of defining automatic differentiation and present different ways of doing automatic differentiation to compute the…
For an associative algebra $A$ a skew-symmetric sum of $n!$ products of $n$ elements of $A$ in all possible order is called $n$-commutator. We consider $A$ as $n$-ary algebra under $n$-commutator. We prove that it has an identity of…
Although in general there is no meaningful concept of factorization in fields, that in free associative algebras (over a commutative field) can be extended to their respective free field (universal field of fractions) on the level 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…
For a couple of associative algebras we define the notion of their double and give a set of examples. Also, we discuss applications of such doubles to representation theory of certain quantum algebras and to a new type of Noncommutative…
In this paper, the definitions of algebras of quotients and Martandale-like qoutients of Leibniz algebras are introduced and the interactions between the two quotients are determined. Firstly, some important properties which not only hold…
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…
In this note we study associative dialgebras proving that the most interesting such structures arise precisely when the algebra is not semiprime. In fact the presence of some "perfection" property (simpleness, primitiveness, primeness or…
This work adapts the equivalent definitions of division algebras over a field into multiple types of division algebras in a monoidal category. Examples and consequences of these definitions are then established in various monoidal settings.
The normal form theorem, proved in R. Laver, On the left distributive law and the freeness of an algebra of elementary embeddings, Advances in Mathematics 91 (1992), 209-231, for the free algebra $\Cal A$ on one generator $x$ satisfying the…
All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classification of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of…
We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.
A free differential for an arbitrary associative algebra is defined as a differential with a uniqueness property. The existence problem for such a differential is posed. The notion of optimal calculi for given commutation rules is…
In this paper, we introduce the notion of algebras of quotients of Hom-Lie algebras and investigate some properties which can be lifted from a Hom-Lie algebra to its algebra of quotients. We also give some necessary and sufficient…
For associative commutative algebras $A$ with Rota-Baxter operator $R$ identities of the algebra $AR=(A,\circ)$, where $a\circ b= aR(b),$ are found.
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…
The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, is shown to be multiplicative for appropriate choices of acceptance windows. This leads to the definition of an associative additive…