<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"].
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