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Keywords
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ideal topology, integral closure, quasiunmixed ring, quintasymptotic prime, regular ring, symbolic power
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Abstract
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Let R be a commutative Noetherian ring and I an ideal of R. The purpose of this paper is to show that
the topologies defined by the integral filtration fI mgm1 and the symbolic integral filtration fI hmigm1 are
equivalent whenever Q.I / consists all of the minimal prime ideals of I . As an application of this result,
by using the Jacobian theorem of Lipman and Sathaye we deduce that the symbolic integral topology
fI hmigm1 is equivalent to the I -adic topology whenever R is a regular ring. Also, applying these results
we provide extensions of classical results of Hartshorne and Zariski on the equivalence of symbolic and
adic topologies.
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