@Suvrit Thank you for the link, that is much closer to the spirit I intended than Stafn's link.

To be more explicit about the mathematics involved, computer-algorithms for calculating images and kernels (and a host of other things) over commutative rings all use Groebner Bases techniques. However, for an easier necessary condition, if $gf=0$, you can tensor with the field of fractions of $R$ and do linear algebra. This cannot prove exactness, but it will quickly eliminate things that aren't exact.