Levels of cancellation for monoids and modules

Levels of cancellativity in commutative monoids $M$, determined by stable rank values in ${\mathbb{Z}}_{>0}\cup \{\mathrm{\infty }\}$ for elements of $M$, are investigated. The behavior of the stable ranks of multiples $ka$, for $k\in {\mathbb{Z}}_{>0}$ and $a\in M$, is determined. In the case of a refinement monoid $M$, the possible stable rank values in archimedean components of $M$ are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.