Consider the set of infinitely generated ideals in R, and show that the union of an ascending chain of ideals in this set cannot be finitely generated. Those generators would appear in one of the ideals, and ideals could not increase beyond this point.
Use zorn's lemma to establish a maximal infinitely generated ideal. Such an ideal is known to be prime, hence it is finitely generated after all.
Any infinitely generated ideal seeds an ascending chain, and leads to a maximal infinitely generated prime ideal, which is a contradiction. Therefore all ideals are finitely generated, and the ring is noetherian.