group scheme of prime order p is killed by p

Question

In the article "Group Schemes of Prime Order" by Tate and Oort (see here) it is proved that a group scheme of prime order $p$ over the base $S$ is killed by $p$ (Theorem 1). The authors state that it clearly suffices to consider the case $S = $ Spec$(R)$, where $R$ is a local ring with algebraically closed residue class field. I don't quite understand this reduction and would be thankful if someone could explain it to me.