A characterization of GCU rings with the FC2 property

We study linear systems (F, G) with coefficients in a commutative ring R with the GCU property: whenever (F, G) is controllable, there exists a vector u with Gu unimodular. Our main result is: a GCU ring is an FC2 ring (i.e. every two-dimensional controllable system over R is feedback cyclizable) if and only if R satisfies the following property: for any finitely generated ideal I and f invertible modulo I there exist a unit u and an element h with f ≡ uh2 (mod I).