Its not the case that every UFD is noetherian; the standard counterexample is $R[x_0,x_1,x_2,\ldots]$, which has the following ascending sequence of ideals:
$$\langle \rangle,\langle x_0\rangle,\langle x_0,x_1\rangle,\langle x_0,x_1,x_2\rangle,\ldots$$
But notice that most of these ideals aren't principal. So:
Question. Is it the case that every UFD is noetherian on principal ideals?