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.



spurious torsion under compositions of linear maps
Here is an obvious way to construct examples with $a=1,b=2$. By elementary divisor theory, with these numerics one always has $f=uf\_0,g=vg\_0$ with $c(f\_0),c(g\_0)=1$ for some $u,v \in R$, so it suffices to study the situation with $f,g$ replaced by $f\_0,g\_0$, i.e. to assume that $c(f),c(g)=1$.


Take $w \in R$ arbitrary. Then $f(x)=(x,0)$ up to choice of basis, and $g(y,z) := wy+z$ satisfies all the conditions, but has $c(g \circ f) = w$.


