Depth One Homogeneous Prime Ideals in Polynomial Rings over a Field

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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/2023/28143

Keywords:

Ideal Basis, Polynomial Ring, Prime Ideal.

Abstract

This paper concerns the question: Which depth one homogeneous prime ideals N in a polynomial ring H are of the principal class? In answer to this question, we introduce acceptable bases of ideals in polynomial rings, and then use a known one-to-one correspondence between the ideals N in H := F[X1, . . . , Xn] such that Xn ∉ N and the maximal ideals P in the related polynomial ring G := F[X1/Xn, . . . , Xn−1/Xn] to show that the acceptable bases of the maximal ideals P in G transform to homogeneous bases. This is used to determine several necessary and sufficient conditions for a given depth one homogeneous prime ideal N in H to be an ideal of the principal class, thus answering, in part, our main question. Then it is shown that the Groebner-grevlex bases of ideals are acceptable bases. Finally, we construct several examples to illustrate our results, and we delve deeper into an example first studied by Macaulay.

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Published

2023-07-12

How to Cite

Kemp, P., Ratliff, L. J., & Shah, K. (2023). Depth One Homogeneous Prime Ideals in Polynomial Rings over a Field. The Journal of the Indian Mathematical Society, 90(3-4), 213–232. https://doi.org/10.18311/jims/2023/28143

Issue

Volume 90, Issue 3-4, July-December 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 2021-07-12
Accepted 2022-07-16
Published 2023-07-12

 

References

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