Let $R$ be a principal ideal ring and let $I$ and $J$ be two ideals of $R$. Suppose $\phi: R \times R \rightarrow R \times R$ and $\psi: R \times R \rightarrow R \times R$ are two $R$-module homomorphisms. Suppose further that $\ker(\phi) = I \times <0>$ and $\ker(\psi) = <0> \times J$.
Then $\psi \circ \phi$ is an $R$-module homomorphism defined on $R \times R$. What can we say about the kernel of $\psi \circ \phi$? Under what conditions will it happen than $\ker(\psi \circ \phi) = I \times J$? Will this be dependent on the map? Any over-arching theorems dealing with this sort of thing?
I have a similar type question here. Any sources other than Dummit and Foote would be good too. (They spend too much time on modules over PID's which doesn't help me.)