Say we have a PID $R$, integers $1 \leq a \leq b$, and $R$-homomorphisms $R^a \stackrel f\to R^b \stackrel g\to R^a$ with $g \circ f$ of full rank.

For $h = f, g, g \circ f$, let $c(h)$ be the characteristic ideal of $\mathrm{coker}(h)_\mathrm{tors}$, i.e. the index of the image of $h$ in its saturation. (In particular $c(g \circ f) = \det(g \circ f)R$, though this interpretation isn't available for $c(f),c(g)$ when $a < b$.)

Keeping $c(f),c(g)$ fixed, can I arrange $c(g \circ f)$ to be divisible by whatever primes of $R$ I want, beyond to those forced to appear by $c(f),c(g)$? If so, I would like a obvious way of constructing examples.