  We study when $R \to S$ has the property that prime ideals of $R$ extend to
prime ideals or the unit ideal of $S$, and the situation where this property
continues to hold after adjoining the same indeterminates to both rings. We
prove that if $R$ is reduced, every maximal ideal of $R$ contains only finitely
many minimal primes of $R$, and prime ideals of $R[X_1,\dots,X_n]$ extend to
prime ideals of $S[X_1,\dots,X_n]$ for all $n$, then $S$ is flat over $R$. We
give a counterexample to flatness over a reduced quasilocal ring $R$ with
infinitely many minimal primes by constructing a non-flat $R$-module $M$ such
that $M = PM$ for every minimal prime $P$ of $R$. We study the notion of
intersection flatness and use it to prove that in certain graded cases it
suffices to examine just one closed fiber to prove the stable prime extension
property.
