  Let $\L$ be a non-noetherian Krull domain which is the inverse limit of
noetherian Krull domains $\L_d$ and let $M$ be a finitely generated $\L$-module
which is the inverse limit of $\L_d$-modules $M_d\,$. Under certain hypotheses
on the rings $\L_d$ and on the modules $M_d\,$, we define a pro-characteristic
ideal for $M$ in $\L$, which should play the role of the usual characteristic
ideals for finitely generated modules over noetherian Krull domains. We apply
this to the study of Iwasawa modules (in particular of class groups) in a
non-noetherian Iwasawa algebra $\Z_p[[\Gal(\calf/F)]]$, where $F$ is a function
field of characteristic $p$ and $\Gal(\calf/F)\simeq\Z_p^\infty$.
