The asymptotic theory of ideals originated with the investigations in a Noetherian ring A of the Samuel number associated with each pair  of non-nilpotent ideals having the same radical and    the limit being reached from below and    It is well known that for all filtrations  and  on a semi-ring A, the sequence  is convergent in  where  is the filtration  and  Then the number  has been introduced. By considering the filtration  we give another extension of  We show, in the case where f is an AP-filtration, that the sequence  is convergent in  Then we put    We get the following result:

ring, semi-ring, filtration, pseudo-valuation