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
is called associative algebra if the following associative law applies to all of them :
So it is a special algebra over a field or a special algebra over a commutative ring .
Examples
- The set of all polynomials with coefficients from a field form (with the usual multiplication) an associative algebra over this field.
- 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.
- 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.
- 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.
- 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.
literature
- Serge Lang : Algebra. Revised 3rd Edition, Springer-Verlag, 2002, ISBN 0-387-95385-X