Abnormalities in Noetherian rings

Authors:
J. T. Arnold and M. B. Boisen

Journal:
Proc. Amer. Math. Soc. **73** (1979), 1-6

MSC:
Primary 13E05; Secondary 13C15

MathSciNet review:
512046

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Abstract: If are prime ideals in some ring *R* and if rank , then is said to be *k*-abnormal and *k* is called the degree of abnormality. The paper consists of two examples. The first example is a Noetherian integral domain in which the set of degrees of abnormality is unbounded. Let *P* be a prime ideal of *R* and set is a prime ideal and is abnormal}. The second example is a local domain such that is *k*-abnormal for some is *k*-abnormal for some *Q* minimal in *W*}.

**[1]**Robert Gilmer,*Multiplicative ideal theory*, Marcel Dekker, Inc., New York, 1972. Pure and Applied Mathematics, No. 12. MR**0427289****[2]**Evan G. Houston Jr. and Stephen McAdam,*Rank in Noetherian rings*, J. Algebra**37**(1975), no. 1, 64–73. MR**0414540****[3]**Irving Kaplansky,*Commutative rings*, Allyn and Bacon, Inc., Boston, Mass., 1970. MR**0254021****[4]**Masayoshi Nagata,*Local rings*, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR**0155856****[5]**L. J. Ratliff Jr.,*Characterizations of catenary rings*, Amer. J. Math.**93**(1971), 1070–1108. MR**0297752**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0512046-2

Keywords:
Prime ideal,
Noetherian ring,
local ring,
rank

Article copyright:
© Copyright 1979
American Mathematical Society