Introduction.

In this section we again restrict attention to noether filtrations which take only non-negative values on a ring A, take some finite positive value, thus satisfying f(1) = 0, and we will be concerned with the valuations v which are associated with such filtrations. We will refer to such valuations as the ideal valuations of A. To begin with we will consider some theorems which enable us to place restrictions on the filtrations we consider and on the ring A. We first deal with the latter. If A has minimal prime ideals Pl, …, Ph, then we have already seen in Theorem 4.16 that the valuations associated with a noether filtration f on A are obtained by lifting the valuations associated with the filtrations f/pj, which are valuations on the fields of fractions of A/pj, to A. This may give rise to a degenerate valuation if radf is contained in pj. Excluding the degenerate valuations, the valuations associated with noether filtrations on A arise from those associated with noether filtrations on the noetherian domains A/pj. Hence, where convenient, we will restrict A to be a noetherian domain, but, where such a restriction is made, it will be stated explicitly.