  Many classical ring-theoretic results state that an ideal that is maximal
with respect to satisfying a special property must be prime. We present a
"Prime Ideal Principle" that gives a uniform method of proving such facts,
generalizing the Prime Ideal Principle for commutative rings due to T.Y. Lam
and the author. Old and new "maximal implies prime" results are presented, with
results touching on annihilator ideals, polynomial identity rings, the
Artin-Rees property, Dedekind-finite rings, principal ideals generated by
normal elements, strongly noetherian algebras, and just infinite algebras.
