Boundness in almost Dedekind domains

In this paper, we study different forms 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 non-critical maximal ideal and, indeed, the set of non-critical 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 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.