A realization theorem for almost Dedekind domains

An integral domain D is called an SP-domain if every ideal is a product of radical ideals. Such domains are always almost Dedekind domains, but not every almost Dedekind domain is an SP-domain. The SP-rank of D provides a natural measure of the deviation of D from being an SP-domain. In the present paper we show that every ordinal number (Formula presented.) can be realized as the SP-rank of an almost Dedekind domain.