Chain of prime ideals in formal power series rings

de S. Doering, Ada Maria; Lequain, Yves

doi:10.1090/S0002-9939-1983-0702281-3

Chain of prime ideals in formal power series rings
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by Ada Maria de S. Doering and Yves Lequain PDF

Proc. Amer. Math. Soc. 88 (1983), 591-594 Request permission

Abstract:

Let $R$ be a Noetherian domain and $P$ a prime ideal of $R$. Then ${R_p}[[{X_1}, \ldots ,{X_n}]]$ has a maximal chain of prime ideals of length $r$ if and only if $R{[[{X_1}, \ldots ,{X_n}]]_{(P,{X_1}, \ldots ,{X_n})}}$ does, if and only if $R{[[{X_1}, \ldots ,{X_n}]]_{(P,{X_1}, \ldots ,{X_n})}}$ does.

References

Jimmy T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973), 299–304. MR 316451, DOI 10.1090/S0002-9947-1973-0316451-8
Jimmy T. Arnold, Algebraic extensions of power series rings, Trans. Amer. Math. Soc. 267 (1981), no. 1, 95–110. MR 621975, DOI 10.1090/S0002-9947-1981-0621975-4
J. T. Arnold and D. W. Boyd, Transcendence degree in power series rings, J. Algebra 57 (1979), no. 1, 180–195. MR 533108, DOI 10.1016/0021-8693(79)90216-3
Yves Lequain, Catenarian property in a domain of formal power series, J. Algebra 65 (1980), no. 1, 110–117. MR 578799, DOI 10.1016/0021-8693(80)90242-2
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. and S. McAdam, Maximal chains of prime ideals in integral extension domains. I, Trans. Amer. Math. Soc. 224 (1976), 103–116. MR 437513, DOI 10.1090/S0002-9947-1976-0437513-3
Philip B. Sheldon, How changing $D[[x]]$ changes its quotient field, Trans. Amer. Math. Soc. 159 (1971), 223–244. MR 279092, DOI 10.1090/S0002-9947-1971-0279092-5