Local Nullstellensatz over Commutative Ground Rings

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Authors

  • Paula Kemp
     Department of Mathematics, Missouri State University, Springfield, Missouri 65897 ,US
  • Louis J. Ratliff
     Department of Mathematics, University of California, Riverside, California 92521-0135 ,US
  • Kishor Shah
     Department of Mathematics, Missouri State University, Springfield, Missouri 65897 ,US

DOI:

https://doi.org/10.18311/jims/2023/28057

Keywords:

G-Ideal, Nullstellensatz, Maximal Ideal, Polynomial Ring.
2010 Mathematics Subject Classification, Primary, 13B25, 13F20, Secondary, 13A15.

Abstract

It is shown that a local Nullstellensatz holds over an arbitrary commutative ring A (with identity 1 ≠ 0); specifically, if B = A[x1, . . . , xn] is a finitely generated extension ring of A and N is a maximal ideal in B, then NBN = (N ∩ A, x1 − c1, . . . , xn − cn)BN for some c1, . . . , cn ∈ BN .

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Published

2023-03-24

How to Cite

Kemp, P., Ratliff, L. J., & Shah, K. (2023). Local Nullstellensatz over Commutative Ground Rings. The Journal of the Indian Mathematical Society, 90(1-2), 149–158. https://doi.org/10.18311/jims/2023/28057

Issue

Volume 90, Issue 1-2, January-June 2023

Section

Articles

License

Copyright (c) 2023 Paula Kemp, Louis J. Ratliff, Kishor Shah

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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Europe PMC
Received 2021-07-01
Accepted 2021-12-26
Published 2023-03-24

 

References

D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995. DOI: https://doi.org/10.1007/978-1-4612-5350-1

I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.

P. Kemp, L. J. Ratliff, Jr., and K. Shah, Depth one homogeneous prime ideals in polynomial rings over a field, Journal of Indian Math. Soc. (accepted).

M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.

O. Zariski and P. Samuel, Commutative Algebra, Vol. 1, D. Van Nostrand, New York, 1958.

O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, D. Van Nostrand, New York, 1960. DOI: https://doi.org/10.1007/978-3-662-29244-0