Associative algebra

Associative algebra is a term from abstract algebra, a branch of mathematics . It is an algebraic structure that expands the concept of vector space or the module in such a way that, in addition to vector addition, an associative multiplication is defined as an inner link .

definition

A vector space A over a body K or a module A over a ring R together with a bilinear map

{\ displaystyle * \ colon A \ times A \ longrightarrow A, \ quad (a, b) \ longmapsto a * b}

is called associative algebra if the following associative law applies to all of them : ${\ displaystyle a, b, c \ in A}$

{\ displaystyle a * (b * c) = (a * b) * c.}

The set of all polynomials with coefficients from a field form (with the usual multiplication) an associative algebra over this field. ${\ displaystyle K}$
The endomorphisms of a vector space form an associative algebra with the concatenation. The link is not commutative, provided the dimension is greater than 1. ${\ displaystyle V}$ ${\ displaystyle *}$ ${\ displaystyle V}$
If an infinite-dimensional vector space is considered and only the endomorphisms with a finite-dimensional image are considered, an example is obtained in which there is no one element. ${\ displaystyle V}$ ${\ displaystyle *}$
The vector space of all real- or complex-valued functions on any topological space forms an associative algebra; the functions are added and multiplied point by point.
The vector space of all continuous real or complex-valued functions on a Banach space forms an associative algebra or even a Banach algebra .
The matrix space of all - matrices together with the matrix multiplication form an associative algebra. ${\ displaystyle (n \ times n)}$

The complex numbers form an associative algebra over the field of real numbers.

The quaternions are an associative algebra over the field of real numbers, but not over the complex numbers.

Serge Lang : Algebra. Revised 3rd Edition, Springer-Verlag, 2002, ISBN 0-387-95385-X