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This is a continuation of a previous paper by the same authors. In the formerpaper, it was proved that in order to obtain local uniformization forvaluations centered on local domains, it is enough to prove it for rank onevaluations. In this paper, we extend this result to the case of valuationscentered on rings which are not necessarily integral domains and may evencontain nilpotents.

Reduction of local uniformization to the case of rank one valuations for rings with zero divisors