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### J. Ratliff, Louis

- Depth One Homogeneous Prime Ideals in Polynomial Rings over a Field

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1 Department of Mathematics, Missouri State University, Springfield, Missouri 65897, US

2 Department of Mathematics, University of California, Riverside, California 92521-0135, US

#### Authors

**Affiliations**

1 Department of Mathematics, Missouri State University, Springfield, Missouri 65897, US

2 Department of Mathematics, University of California, Riverside, California 92521-0135, US

#### Source

The Journal of the Indian Mathematical Society, Vol 90, No 3-4 (2023), Pagination: 213–232#### 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.#### Keywords

Ideal Basis, Polynomial Ring, Prime Ideal.#### References

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