  We study different form of boundness for ideals of almost Dedekind domains,
generalizing the notions of critical ideals, radical factorization, and
SP-domains. We show that every almost Dedekind domain has at least one
noncritical maximal ideals and, indeed, the set of noncritical maximal ideals
is dense in the maximal space, with respect to the constructible topology; as a
consequence, we show that every almost Dedekind domain is SP-scattered, and in
particular that the group $\mathrm{Inv}(D)$ of invertible ideals of an almost
Dedekind domain $D$ is always free. If $D$ is an almost Dedekind domain with
nonzero Jacobson radical, we also show that there is at least one element whose
ideal function is bounded.
