Let O be any prime of U . Then the strong image of O in each X Q is a prime containg its minimal prime. Thus the strong image of O in X contains each Q . By Zorn's lemma, S has a maximal element, which is a maximal prime object containing P . Suppose P is a maximal prime of X .
X be the localization of X at P .
Xp be the pullback of f and i . It follows that i ' is non-initial. Let Q be any prime of T . One direction is clear. Suppose X is a von Neumann regular object.