I would appreciate some help with developing a simple proof that the ideals in the ring of integers for a number field have the same rank as the ring of integers itself. In other words, assuming from the start that all the ideals are finitely generated, all ideals require the same number of generators as the entire ring of integers itself. I find it easy to show that the ideals have the same rank or lower, but not that they have to have the same rank. Any help would be appreciated. Thanks.