Let υ be a prime divisor of a 2-dimensional regular local ring (Rm) with algebraically closed residue fieldk. Zariski showed that a prime divisor υ ofRis uniquely associated to a simplem-primary integrally closed idealIofR, there exist finitely many simple υ-ideals includingI, and all the other υ-ideals can be uniquely factored into products of simple υ-ideals. It is known that such anm-primary idealIofRcan be minimally generated by o(I) + 1 elements.Given a simple integrally closed idealIof order one with arbitrary rank and its associated prime divisor υ, we find minimal generating sets of all the simple υ-ideals and describe factorizations of all the composite υ-ideals in terms of power products of simple υideals as explicitly as possible.