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

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

MathSciNet review:
512046

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Abstract | References | Similar Articles | Additional Information

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]**R. Gilmer,*Multiplicative ideal theory*, Marcel Dekker, New York, 1972. MR**0427289 (55:323)****[2]**E. G. Houston and S. McAdam,*Rank in Noetherian rings*, J. Algebra**37**(1975), 64-73. MR**0414540 (54:2641)****[3]**I. Kaplansky,*Commutative rings*, Allyn and Bacon, Boston, Mass., 1970. MR**0254021 (40:7234)****[4]**M. Nagata,*Local rings*, Interscience, New York, 1962. MR**0155856 (27:5790)****[5]**L. J. Ratliff, Jr.,*Characterizations of catenary rings*, Amer. J. Math.**93**(1971), 1070-1108. MR**0297752 (45:6804)**

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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