Unital

In mathematics, an associative algebra is unital if it contains a multiplicative identity element (or unit ), i.e. an element 1 with the property 1 x = x 1 = x for all elements x of the algebra.

This is equivalent to say 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 and Matrix (mathematics), 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 s.

According to the glossary of ring theory, the Wikipedia convention assumes the existence of a multiplicative identity for any ring (mathematics). With this assumption, all rings are unital, and all ring homomorphisms are unital, and (associative) algebras are unital iff they are rings.