Gauge Functions and the Divisor Chain Condition

A gauge function on a commutative cancellative monoid M is a map p : M→Z+1 (the set of nonnegative) integers such that for all a,b ε M, p(ab)≥ p(a)+p(b) and p(a) = 0 if and only if a is a unit. In [3], it is shown by V. Srinivasan and H. Shaing that if the monoid of nonzero elements of an integral domain A in which finitely generated ideals are principal admits a gauge function, then A is a principal ideal domain. In this paper I show that M admits a gauge function if and only if for each nonunit a ε M there is a positive integer N = N(a) such that α is a product of no more than N(a) irreducibles. An example is given to show that this latter condition is stronger than the divisor chain condition defined in [2, 2.14]