 The integers are one-dimensional, since chains are of the form (0) ⊊ ("p"), where "p" is a prime number.For non-Noetherian rings, and also non-local rings, the dimension may be infinite, but Noetherian local rings have finite dimension.Among the four axioms above, the first two are elementary consequences of the definition, whereas the remaining two hinge on important facts in commutative algebra, the going-up theorem and Krull's principal ideal theorem.