Associative algebra

Associative algebra is a term used in abstract algebra a branch of mathematics. Is an algebraic structure, the effect extends the concept of a vector space, or the module, that in addition to the vector addition of an associative multiplication is defined as the inner link.

Definition

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

Called associative algebra if the following associative law applies to all:

It is a special algebra over a field or a special algebra over a commutative ring.

Examples

• Forming the set of polynomials with coefficients in a body ( with the usual multiplication) an associative algebra over that body.

• The endomorphisms of a vector space form an associative algebra with the concatenation. Here, the link is not commutative unless the dimension of greater than 1.

• If an infinite-dimensional vector space, and only considering the endomorphisms with finite- dimensional range, you get an example in which has no identity element.

• The vector space of all real - or complex-valued functions on any topological space forms an associative algebra; while the functions are pointwise added and multiplied.

• The vector space of all continuous real - or complex-valued functions on a Banach space is an associative algebra, or even a Banach algebra.

• The die space of all matrices together with matrix multiplication is 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.