Algebra

The term algebra is used in mathematics in several different senses.

At an elementary level, algebra involves the manipulation of simple equations in real (or sometimes complex) variables. See Elementary algebra.

More generally, algebra (or abstract algebra) is the study of algebraic structures such as groups, rings and fields. See Abstract algebra for further details.

An algebra over a field (or simply an algebra) is a vector space together with a vector multiplication that distributes over vector addition and has the further property that (ax)(by) = (ab)(xy) for all scalars a and b and all vectors x and y. For example, a field is an algebra over any of its subfields, and the quaternions, octonions and sedenions are algebras over the real numbers. Another example is R³ with the usual 3-dimensional vector multiplication.

See also Boolean algebra, sigma-algebra? and linear algebra.