Integral degree of a ring, reduction numbers and uniform Artin–Rees numbers

The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains bounds for the Castelnuovo–Mumford regularity of the Rees algebra and for the Artin–Rees numbers.