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The Laskerian property, power series rings and Noetherian spectra

Authors: Robert Gilmer and William Heinzer
Journal: Proc. Amer. Math. Soc. 79 (1980), 13-16
MSC: Primary 13E05; Secondary 13J05
MathSciNet review: 560575
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Abstract: We show that if the power series ring $ R[[X]]$ in one indeterminate over a commutative ring R with identity is Laskerian, then R is Noetherian. On the other hand, if $ R[[X]]$ is a ZD-ring, then R has Noetherian spectrum, but R need not be Noetherian. We show that, in general, a Laskerian ring has Noetherian spectrum.

References [Enhancements On Off] (What's this?)

  • [AGH] J. T. Arnold, R. Gilmer and W. Heinzer, Some countability conditions in a commutative ring, Illinois J. Math. 21 (1977), 648-665. MR 0460316 (57:310)
  • [B] N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., 1972.
  • [E] E. G. Evans, Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155 (1971), 505-512. MR 42 #7654. MR 0272773 (42:7654)
  • [HO] W. Heinzer and J. Ohm, On the Noetherian-like rings of E. G. Evans, Proc. Amer. Math. Soc. 34 (1972), 73-74. MR 45 #3385. MR 0294316 (45:3385)
  • [M] S. Mori, Über eindeutige Reduktion in Ringen ohne Teilerkettensatz, J. Sci. Hiroshima Univ. Ser. A 3 (1933), 275-318.
  • [OP] J. Ohm and R. Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968), 631-640. MR 37 #5201. MR 0229627 (37:5201)
  • [ZS] O. Zariski and P. Samuel, Commutative algebra, vol. I, The University Series in Higher Mathematics, Van Nostrand, Princeton, N. J., 1958. MR 0090581 (19:833e)

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Keywords: Laskerian ring, power series ring, Noetherian, ZD-ring, Noetherian spectrum
Article copyright: © Copyright 1980 American Mathematical Society

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