  We study decompositions of length functions on integral domains as sums of
length functions constructed from overrings. We find a standard representation
when the integral domain admits a Jaffard family, when it is Noetherian and
when it is a Pr\"ufer domains such that every ideal has only finitely many
minimal primes. We also show that there is a natural bijective correspondence
between singular length functions and localizing systems.

Commutative Algebra;Rings and Algebras