Hopf Algebra

Let a graded module $A$ have a multiplication $\phi$ and a co-multiplication $\psi$. Then if $\phi$ and $\psi$ have the unity of $k$ as unity and $\psi:(A,\phi)\to(A,\phi)\otimes(A,\phi)$ is an algebra homomorphism, then $(A,\phi,\psi)$ is called a Hopf algebra.