Is the power set P(S) under union and intersection a ring.
If not, why not?

Additive inverses. See if you can find a counterexample for yourself. By that, I mean find a set S and a subset of S with no additive inverse. By that, I mean a subset of S which, when you union it with any other subset of S, you never get the additive identity (in this case, the empty set).