Unital
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In mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative identity element (or unit), i.e. an element 1 with the property 1x = 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 → 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, 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.
