Note on symbolic powers and going down

McAdam, S.; Ratliff, L. J.

doi:10.1090/S0002-9939-1986-0854018-8

Note on symbolic powers and going down
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by S. McAdam and L. J. Ratliff PDF

Proc. Amer. Math. Soc. 98 (1986), 199-204 Request permission

Abstract:

For primes $P \subseteq Q$ of a Noetherian ring $R$, we consider when, for all $k \geq 1$, there is an $m$ with ${P^{(m)}} \subseteq {Q^{(k)}}$ and reprove a relevant theorem of Schenzel. If $R$ is a domain, we consider sufficient conditions for $P \subseteq Q$ to satisfy going down for all primes $Q$ containing $P$.

References

Commutative ring theory

Stephen McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983. MR 722609, DOI 10.1007/BFb0071575
Stephen McAdam, Filtrations, Rees rings, and ideal transforms, J. Pure Appl. Algebra 42 (1986), no. 3, 237–243. MR 857893, DOI 10.1016/0022-4049(86)90009-5
Stephen McAdam and L. J. Ratliff Jr., Essential sequences, J. Algebra 95 (1985), no. 1, 217–235. MR 797664, DOI 10.1016/0021-8693(85)90102-4
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
L. J. Ratliff Jr., The topology determined by the symbolic powers of primary ideals, Comm. Algebra 13 (1985), no. 9, 2073–2104. MR 795491, DOI 10.1080/00927878508823265
Peter Schenzel, Symbolic powers of prime ideals and their topology, Proc. Amer. Math. Soc. 93 (1985), no. 1, 15–20. MR 766518, DOI 10.1090/S0002-9939-1985-0766518-9