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.

(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 boundedlinear 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.