Invariants of ideals having principal reductions

For a regular ideal having a principal reduction in a Noetherian ring we co nsider the structural numbers that arise from taking the Ratliff-Rush closu re of the ideal and its powers. In particular, we analyze the interconnecti ons among these numbers and the relation type and reduction number of the i deal. We prove that certain inequalites hold in general among these invaria nts, while for ideals contained in the conductor of the integral closure of the ring we obtain sharper results that do not hold in general. We provide applications to the one-dimensional local setting and present a sequence o f examples in this context.