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On Nagata’s Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (II)


Affiliations
1 Department of Mathematics, Missouri State University, Springfield, United States
2 Department of Mathematics, University of California, Riverside, United States
     

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We expand the theory of height one maximal ideals and depth one minimal prime ideals initiated by M. Nagata and continued by the authors in part I. A local ring is doho in case its completion has at least one depth one minimal prime ideal. We establish several families of doho local rings, prove that certain local rings associated with Rees valuation rings are doho, and complement a famous construction of Nagata by proving that each doho local domain (<I>R,M</I>) of altitude α ≥ 2 has a quadratic integral extension over-domain with precisely two maximal ideals, one of height α and the other of height one.

Keywords

Completion of a Local Ring, Depth One Minimal Prime Ideal, Height One Maximal Ideal, Rees Valuation Ring.
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  • Paula Kemp, Louis J. Ratliff, Jr., and Kishor Shah, On Nagata’s result about height one maximal ideals and depth one minimal prime ideals (I), J. Indian Math. Soc., (to appear).
  • D. Katz and J. Validashti, Multiplicities and Rees valuations, Collect. Math. 61 (2010), 1-24.
  • M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.
  • L. J. Ratliff, Jr., On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals (II), Amer. J. Math. 92 (1970), 99-144.
  • I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Modules, Cambridge Univ. Press, Cambridge, 2006.

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  • On Nagata’s Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (II)

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Authors

Paula Kemp
Department of Mathematics, Missouri State University, Springfield, United States
Louis J. Ratliff
Department of Mathematics, University of California, Riverside, United States
Kishor Shah
Department of Mathematics, Missouri State University, Springfield, United States

Abstract


We expand the theory of height one maximal ideals and depth one minimal prime ideals initiated by M. Nagata and continued by the authors in part I. A local ring is doho in case its completion has at least one depth one minimal prime ideal. We establish several families of doho local rings, prove that certain local rings associated with Rees valuation rings are doho, and complement a famous construction of Nagata by proving that each doho local domain (<I>R,M</I>) of altitude α ≥ 2 has a quadratic integral extension over-domain with precisely two maximal ideals, one of height α and the other of height one.

Keywords


Completion of a Local Ring, Depth One Minimal Prime Ideal, Height One Maximal Ideal, Rees Valuation Ring.

References