Normalization of rings

We present a new algorithm to compute the integral closure of a reduced Noetherian ring in its total ring of fractions. We proceed, as in the classical case, by constructing an ascending chain of endomorphism rings of test ideals. However, our approach avoids the increasing complexity when enlarging the rings by doing most computations over the original ring. A modification, applicable in positive characteristic, where actually all computations are over the original ring, is also described. The new algorithm of this paper is developed, and has been implemented in Singular, for localizations of affine rings with respect to arbitrary monomial orderings. Benchmark tests show that it is in general much faster than any other algorithm known to us.