Let [Formula: see text] be an odd prime number and [Formula: see text]. Assume that either [Formula: see text]i[Formula: see text] [Formula: see text] or [Formula: see text]ii[Formula: see text] [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text] [Formula: see text]respectively, assume that [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text]. Then there exist exactly five [Formula: see text]respectively, exactly seven[Formula: see text] isomorphism classes of rings [Formula: see text] for which there exists a tower [Formula: see text] of ramified [Formula: see text]integral minimal[Formula: see text] ring extensions such that [Formula: see text] is the only ring properly contained between [Formula: see text] and [Formula: see text]. We produce a set of isomorphism class representatives of such [Formula: see text] and, for each such [Formula: see text], we show that, up to isomorphism, the corresponding ring [Formula: see text] is the idealization [Formula: see text]. One consequence, for each integer [Formula: see text] whose prime-power factorization [Formula: see text] (with pairwise distinct prime numbers [Formula: see text] and positive integers [Formula: see text]) satisfies [Formula: see text] for all [Formula: see text], is a classification, up to isomorphism, of the rings that have cardinality [Formula: see text] and exactly two proper (unital) subrings.
