TITLE: Categorization of Group Scheme of rank 2
QUESTION [2 upvotes]: I got stuck on this Exercise 2.11 in the book Introduction to Affine Group Scheme of Waterhouse. I really appreciate if anyone can give me a hint.

Let $A$ be a Hopf algebra over $k$ (a base ring) which is free of rank 2. I proved that $I = Ker(\epsilon) = kx$ for some $x$, and $\Delta(x) = x \otimes 1 + 1 \otimes x + b x\otimes x$ for some $b$ in k. Show that $x^2 + ax = 0$ for some $a$, so $A = k[X]/(X^2 + aX)$.

I guess we can obtain the equation $x^2 + ax = 0$ from $\Delta(x) = x \otimes 1 + 1 \otimes x + b x\otimes x$. We can apply multiplication to both sides of the equation, but I don't know how $\Delta$ behaves under multiplication. Thanks in advance.

REPLY [1 votes]: Since $\{1, x\}$ is a basis for $A$, you can write $x^2 = c_1 + c_2 x$ for some $c_1, c_2 \in k$.  Apply $\epsilon$ to this equation to show that $c_1 = 0$, so that $x^2 = c_2 x$.
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