Hopf algebra kernels vs. algebra kernels Let $f: H_1 \rightarrow H_2$ be a map of graded connected cocommutative Hopf algebras over a perfect field.  Let $H \subset H_1$ be the Hopf algebra kernel of $f$, and let $I \subset H_1$ be the kernel of $f$, viewed as an algebra map.
Let $\bar H$ be the positive dimensional part of $H$, and let $(\bar H) \subset H_1$ be the algebra ideal generated by $\bar H$.  Clearly $(\bar H) \subseteq I$.
Questions:  Does $(\bar H) = I$?  It seems likely that this is a standard fact. If so, where is this in the literature?   
(The hypotheses that the Hopf algebras are cocommutative, and having the field be perfect, just happen to hold in the situation I am considering, and perhaps are irrelevant.)
 A: The paper "A correspondence between bi-ideals and sub-Hopf algebras in cocommutative Hopf algebras" by K. Newman (J. Algebra, Volume 36, Issue 1, July 1975, Pages 1-15) may answer your question.
See also Susan Montgomery, "Hopf algebras and their actions on rings," in particular the question at the bottom of p. 36, Theorem 3.4.6, and ensuing discussion. (This is what led me to Newman's paper.) And as I said in a comment, Proposition 1.3 in Wilkerson's paper "The Cohomology Algebras of Finite Dimensional Hopf Algebras" may be relevant (and see also Theorem 4.9 in Milnor-Moore), although it is only stated for surjections of Hopf algebras. Wilkerson is the only one of these working in the graded connected setting, so if you have graded connected Hopf algebras which are not actually Hopf algebras if you forget the grading, you should take care with the other results.
A: -too long for a comment-
I am a little confused about the way terminology is used in the OP.
Maybe i'm missing the point; in case i do not, the closest result i know of -quite general and does not refer specifically to graded or connected or cocommutative case- is Lemma 16.0.2, p. 306, of Sweedler's book.
Copying verbatim: 

Let $K$ and $L$ be hopf algebras and $\pi:K\to L$ a surjective hopf algebra map. Let $A=\{ g\in K / (Id\otimes\pi)\Delta(g)=g\otimes 1$. $A$ is a subalgebra of $K$ and $\varepsilon_K|A$ is an augmentation. The augmentation ideal $(\ker\varepsilon)\cap A$ is denoted $A^+$. If $A^+ K$ denotes the right ideal in $K$, generated by $A^+$, we have: $$\ker\pi=A^+K$$ 

