#### Title

Gauge Functions and the Divisor Chain Condition

#### Document Type

Article

#### Department

Mathematics (HMC)

#### Publication Date

9-1983

#### Abstract

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].

#### Rights Information

© 1983 Taylor & Francis Group

#### DOI

10.1080/0020739830140502

#### Recommended Citation

Henriksen, Melvin. 1983. Gauge functions and the divisor chain condition. International Journal of Mathematical Education in Science and Technology. 14(5):551-553.