Given a Hopf algebra $H$, a left $H$-comodule $V$, and a surjective comodule endomorphism $f: V \to V$. Can somebody give:

(i) a set of neccessary, or sufficient, or both neccessary and sufficient, conditions for $f$ to have zero kernel?

(ii) an example of such a comodule map with non-zero kernel?

Thanks in advance guys

socle. – Mariano Suárez-Alvarez♦ Aug 22 '12 at 17:46