Krull dimension in power series rings

Arnold, Jimmy T.

doi:10.1090/S0002-9947-1973-0316451-8

Krull dimension in power series rings
HTML articles powered by AMS MathViewer

by Jimmy T. Arnold

Trans. Amer. Math. Soc. 177 (1973), 299-304

DOI: https://doi.org/10.1090/S0002-9947-1973-0316451-8

PDF | Request permission

Abstract:

Let R denote a commutative ring with identity. If there exists a chain ${P_0} \subset {P_1} \subset \cdots \subset {P_n}$ of $n + 1$ prime ideals of R, where ${P_n} \ne R$, but no such chain of $n + 2$ prime ideals, then we say that R has dimension n. The power series ring $R[[X]]$ may have infinite dimension even though R has finite dimension.

References

J. T. Arnold and J. W. Brewer, When $(D[[X]])_{P[[X]]}$ is a valuation ring, Proc. Amer. Math. Soc. 37 (1973), 326–332. MR 311656, DOI 10.1090/S0002-9939-1973-0311656-X
David E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc. 27 (1971), 427–433. MR 271100, DOI 10.1090/S0002-9939-1971-0271100-6
David E. Fields, Dimension theory in power series rings, Pacific J. Math. 35 (1970), 601–611. MR 277518, DOI 10.2140/pjm.1970.35.601
Robert W. Gilmer, Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR 0229624
Jack Ohm and R. L. Pendleton, Rings with noetherian spectrum, Duke Math. J. 35 (1968), 631–639. MR 229627
A. Seidenberg, A note on the dimension theory of rings, Pacific J. Math. 3 (1953), 505–512. MR 54571, DOI 10.2140/pjm.1953.3.505
A. Seidenberg, On the dimension theory of rings. II, Pacific J. Math. 4 (1954), 603–614. MR 65540, DOI 10.2140/pjm.1954.4.603