On some classes of R-modules and congruences in R-tors

For a regular ideal having a principal reduction in a Noetherian ring we consider the structural numbers that arise from taking the Ratliff–Rush closure of the ideal and its powers. In particular, we analyze the interconnections among these numbers and the relation type and reduction number of the ideal. We prove that certain inequalites hold in general among these invariants, 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 of examples in this context.