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Krull's principal ideal theorem in non-Noetherian settings

Published online by Cambridge University Press:  08 August 2018

BRUCE OLBERDING
BRUCE OLBERDING*
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001, U.S.A. e-mail: [email protected]
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Abstract

Let P be a finitely generated ideal of a commutative ring R. Krull's principal ideal theorem states that if R is Noetherian and P is minimal over a principal ideal of R, then P has height at most one. Straightforward examples show that this assertion fails if R is not Noetherian. We consider what can be asserted in the non-Noetherian case in place of Krull's theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 13 - 27
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

REFERENCES

[1] Anderson, D. D., Matijevic, J. and Nichols, W. The Krull intersection theorem II. Pacific J. Math. 66 (1976), 15–22.Google Scholar
[2] Anderson, D. F., Dobbs, D., Eakin, P. and Heinzer, W. On the generalised principal ideal theorem and Krull domains. Pacific J. Math. 146 (1990), no. 2, 201–215.Google Scholar
[3] Barucci, V., Anderson, D. F. and Dobbs, D. Coherent Mori domains and the principal ideal theorem. Comm. Algebra 15 (1987), no. 6, 1119–1156.Google Scholar
Cahen, P.–J., Houston, E. G and Lucas, T. G. Answer to a question on the principal ideal theorem. Zero-dimensional commutative rings (Knoxville, TN, 1994), 163–166. Lecture Notes in Pure and Appl. Math. 171 (Dekker, New York, 1995).Google Scholar
[5] Cohen, I. S. On the structure and ideal theory of complete local rings. Trans. Amer. Math. Soc. 59, (1946), 54–106.Google Scholar
[6] Eakin, P. and Sathaye, A. Prestable ideals. J. Algebra 41 (1976), no. 2, 439–454.Google Scholar
[7] Gilmer, R., Heinzer, W. and Roitman, M. Finite generation of powers of ideals. Proc. Amer. Math. Soc. 127 (1999), no. 11, 3141–3151.Google Scholar
[8] Gabelli, S. and Roitman, M. Finitely stable domains, II Preprint.Google Scholar
Heinzer, W. and Roitman, M. Generalised local rings and finite generation of powers of ideals, in Non-Noetherian commutative ring theory, Math. Appl., vol. 520, pages 287–312 (Kluwer Acad. Publ., Dordrecht, 2000).Google Scholar
[10] Heinzer, W., Rotthaus, C. and Wiegand, S. Examples using power series over Noetherian integral domains Preprint.Google Scholar
Huckaba, J. Commutative Rings with Zero Divisors (Marcel Dekker, 1988).Google Scholar
[12] Kang, B. G. and Mulay, S. B. A generalised principal ideal theorem. J. Pure Appl. Algebra 211 (2007), no. 1, 51–54.Google Scholar
[13] Knebusch, M. and Zhang, D. Manis valuations and Prüfer extensions. I. A new chapter in commutative algebra. Lecture Notes in Mathematics, 1791 (Springer-Verlag, Berlin, 2002).Google Scholar
[14] Krull, W. Primidealketten in allgemeinen Ringbereichen, Heidelberg, S. B., Akad. Weiss. (1928), 7 Abh.Google Scholar
[15] Matijevic, J. Maximal ideal transforms of Noetherian rings. Proc. Amer. Math. Soc. 54 (1976), 49–52.Google Scholar
[16] Matsumura, H. Commutative Ring Theory (Cambridge University Press, 1986).Google Scholar
[17] Northcott, D. G. Ideal theory. Cambridge Tracts in Mathematics and Mathematical Physics, no. 42 (Cambridge, at the University Press, 1953).Google Scholar
[18] Nagata, M. Local rings. Interscience Tracts in Pure and Applied Mathematics, no. 13 (John Wiley & Sons, New York-London, 1962).Google Scholar
[19] Ohm, J. Some counterexamples related to integral closure in D[[x]]. Trans. Amer. Math. Soc. 122 (1966) 321–333.Google Scholar
[20] Olberding, B. Finitely stable rings. Commutative algebra, 269–291 (Springer, New York, 2014).Google Scholar
[21] Olberding, B. One-dimensional stable rings. J. Algebra 456 (2016), 93–122.Google Scholar
[22] Olberding, B. and Shapiro, J. Prime ideals in ultraproducts of commutative rings. J. Algebra 285 (2005), 768–794.Google Scholar
[23] Sally, J. D. Numbers of Generators of Ideals in Local Rings (Marcel Dekker, Inc., New York-Basel, 1978).Google Scholar
[24] Schoutens, H. Dimension and singularity theory for local rings of finite embedding dimension. J. Algebra 386 (2013), 1–60.Google Scholar
[25] Swanson, I. and Huneke, C. Integral closure of ideals, rings, and modules, London Math. Soc. Lecture Note Series, 336 (Cambridge University Press, Cambridge, 2006).Google Scholar