Algebra
<mathematics, logic> 1. A loose term for an
algebraic structure.
2. A
vector space that is also a ring, where the vector space and the ring share the same addition operation and are related in certain other ways.
An example algebra is the set of 2x2 matrices with real numbers as entries, with the usual operations of addition and matrix multiplication, and the usual
scalar multiplication. Another example is the set of all polynomials with real coefficients, with the usual operations.
In more detail, we have:
(1) an underlying
set,
(2) a
field of scalars,
(3) an operation of scalar multiplication, whose input is a scalar and a member of the underlying set and whose output is a member of the underlying set, just as in a
vector space,
(4) an operation of addition of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a vector space or a ring,
(5) an operation of multiplication of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a ring.
This whole thing constitutes an `algebra' iff:
(1) it is a vector space if you discard item (5) and
(2) it is a ring if you discard (2) and (3) and
(3) for any scalar r and any two members A, B of the underlying set we have r(AB) = (rA)B = A(rB).
In other words it doesn't matter whether you multiply members of the algebra first and then multiply by the scalar, or multiply one of them by the scalar first and then multiply the two members of the algebra.
Note that the A comes before the B because the multiplication is in some cases not commutative, e.g. the matrix example.
Another example (an example of a
Banach algebra) is the set of all
bounded linear operators on a Hilbert space, with the usual
norm.
The multiplication is the operation of
composition of operators, and the addition and scalar multiplication are just what you would expect.
Two other examples are tensor algebras and Clifford algebras.
[I. N. Herstein, "Topics_in_Algebra"].
