Unital Algebra - Unital Homomorphism

Unital Homomorphism

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.