This post is about examples of some nice things that could fail for modules that are not finitely generated (or not finitely presented when the ring is not Noetherian).
Let be a commutative noetherian ring, an ideal of and a finitely generated -module, then we have an isomorphism where the completion is with respect to the -adic topology [III. 3.4 Theorem 3, Commutative Algebra, Bourbaki].
Note that when is not noetherian, being finitely presented would suffice.
Here are two counterexamples showing that the map above need not be injective or surjective when is not finitely generated: https://math.stackexchange.com/q/1143557
Let , is this ring normal (integral closed in its field of fraction)?
Edit (see comments below): Because is of height 1, then there exists one among say such that for any . Otherwise, if there are polynomials in one variable for each of the variables contained in , then is of height . Thus becomes invertible in for any .