Is the dual of a finitely generated module finitely generated?

I recently thought of this and have no idea whether over a general commutative unitary ring the dual of a finitely generated module is finitely generated. This must be known.

If $I$ is an ideal of $R$, the dual of $R/I$ is isomorphic to $\mathrm{Ann}(I) = \{r \in R : rI = 0\}$, and this doesn't have to be finitely generated. Take for instance $R = k[y,x_1,x_2,\dotsc]/(y x_i : i \geq 1)$ and $I=(y)$.

A more natural question would be: How can we characterize commutative rings with the property that duals of f.g. modules over that ring are f.g.? Even more natural: How can we characterize commutative rings with the property that hom modules between f.g. modules are f.g.? For example, noetherian commutative rings satisfy this property (see the comment by Keenan Kidwell). But I think that there are more examples (perhaps coherent rings?).