Defintion |
Definition |
The dimension of the associative algebra A over the field K is its dimension as a K-vector space. |
* The complex numbers form a unitary associative algebra over the real numbers, and so do the quaternions. |
* The complex numbers form a 2-dimensional unitary associative algebra over the real numbers * The quaternions form a 4-dimensional unitary associative algebra over the reals and a 2-dimensional unitary associative algebra over the complex numbers. |
The dimension of the associative algebra A over the field K is its dimension as a K-vector space.
If A and B are associative algebras over the same field K, an algebra homomorphism f : A -> B is a K-linear map which is also multiplicative in the sense that f(xy) = f(x) f(y) for all x, y in A. With this notion of morphism, the class of all associative algebras over K becomes a category.
Take for example the algebra A of all real-valued continuous functions R -> R, and B = R. Both are algebras over R, and the map which assigns to every continuous function φ the number φ(0) is an algebra homomorphism from A to B.
One may consider associative algebras over a commutative ring R: these are modules over R together with a R-bilinear map which yields an associative multiplication.
The n-by-n matrices with integer entries form an associative algebra over the integers and the polynomials with coefficients in the ring Zn (see modular arithmetic) form an associative algebra over Zn.