The types of local rings that arise, however, depend closely on the theory under consideration. The Weierstrass preparation theorem implies that rings of germs of holomorphic functions are Noetherian rings. It can also be shown that these are regular rings. On the other hand, let formula_69 be the ring of germs at the origin of smooth functions on R. This ring is local but not Noetherian. To see why, observe that the maximal ideal "m" of this ring consists of all germs that vanish at the origin, and the power "m""k" consists of those germs whose first "k" − 1 derivatives vanish. If this ring were Noetherian, then the Krull intersection theorem would imply that a smooth function whose Taylor series vanished would be the zero function. But this is false, as can be seen by considering