In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i.e. an element 1 with the property 1x = x1 = x for all elements x of the algebra.

Such a multiplicative identity element, if it exists, is unique.

Most associative algebras considered in abstract algebra, for instance group algebras, polynomial algebras and matrix algebras, are unital. Some algebras that naturally arise in analysis are not unital, for instance the algebra of functions 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 AxK 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 AxK 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: all our rings are unital, and all our ring homomorphisms are unital. This convention is not universal in the mathematical literature.