If "Z" ⊂ "X" is a prime divisor, then the local ring formula_6 has Krull dimension one. If formula_7 is non-zero, then the order of vanishing of "f" along "Z", written , is the length of formula_8 This length is finite, and it is additive with respect to multiplication, that is, . If "k"("X") is the field of rational functions on "X", then any non-zero may be written as a quotient , where "g" and "h" are in formula_9 and the order of vanishing of "f" is defined to be . With this definition, the order of vanishing is a function . If "X" is normal, then the local ring formula_6 is a discrete valuation ring, and the function is the corresponding valuation. For a non-zero rational function "f" on "X", the principal Weil divisor associated to "f" is defined to be the Weil divisor