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Let I be an ideal of a Noetherian ring R and let S be a multiplicatively closed subset of R. The purpose of this paper is to study the relationship between some properties of the associated graded ring Gr(I, R), and the comparison of the topologies defined by the I-adic filtration {In}n≥=0, the integral filtration , and the (S)-symbolic filtration {S(In)}n≥=0.
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