  Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism
$\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when
the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf
algebra structure that is compatible with that of $R$. In fact, they gave a
complete characterization of which $\sigma$ and $\delta$ can occur under the
hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a,
b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the
comultiplication map. In this paper, we show that after a change of variables
one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$,
with $\beta $ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when
$R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this
completely characterizes skew polynomial extensions of a Hopf algebra that
admit a Hopf structure extending that of the ring of coefficients under these
hypotheses. We show that the hypotheses hold for domains $R$ that are
noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.
