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The topology determined by the symbolic powers of primary ideals

Let q be a p-primary ideal in a Noetherian ring R. The main theorem characterizes when the q-adic and q-symbolic topologies on R are linearly equivalent; that is, when there n kexists an integer k≥0 such that q(n) ⊆qn-k for all n≥k. Using this, it is shown that when this holds for q, then it holds for several other primary ideals related to q (both in R and in certain other rings related to R) and that the powers of q(n) are symbolic powers for all n≥k (so p has primary ideals all of whose powers are symbolic powers).