Unital
In
mathematics, an algebra is
unital (some authors use
unitary) if it contains a multiplicative
identity element (or
unit), i.e. an element 1 with the property 1
x =
x1 =
x for all elements
x of the algebra.
This is equivalent to saying that the algebra is a
monoid for multiplication. As in any monoid, such a multiplicative identity element is then unique.
Most
associative algebras considered in
abstract algebra, for instance
group algebras,
polynomial algebras and
matrix algebras, are unital, if rings are assumed to be so.Most algebras of functions considered in
analysis are not unital, for instance the algebra of square integrable functions (defined on an unbounded domain), and the algebra of functions decreasing to zero at infinity, especially those with
compact support on some (non-
compact) space.
Given two unital algebras
A and
B, an algebra
homomorphism f :
A â†'
B is
unital if it maps the identity element of
A to the identity element of
B.
If the associative algebra
A over the
field K is
not unital, one can adjoin an identity element as follows: take
A×
K as underlying
K-
vector space and define multiplication * by (
x,
r) * (
y,
s) = (
xy +
sx +
ry,
rs) for
x,
y in
A and
r,
s in
K. Then * is an associative operation with identity element (0,1). The old algebra
A is contained in the new one, and in fact
A×
K is the "most general" unital algebra containing
A, in the sense of
universal constructions.
According to the
glossary of ring theory, the Wikipedia convention assumes the existence of a multiplicative identity for any
ring.With this assumption, all rings are unital, and all ring homomorphisms are unital, and (associative) algebras are unital
iff they are rings. Authors who do not require rings to have identity will refer to rings which do have identity as unital rings, and
modules over these rings for which the ring identity acts as an identity on the module as unital modules or unitary modules.
