Abstract Views :172 |
PDF Views:3
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
- W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Math., 39, Cambridge Univ. Press, First paperback ed. with revisions, 1998.
- R. C. Cowsik and M. V. Nori, On the fibres of blowing up, J. Indian Math. Soc. 40(1976), 217-222.
- David Cox, John Little, Donal O’Shea, Ideals, Varieties, and Algorithms, 4th Ed., Springer, New York 2015.
- N. Jacobson, Lectures in Abstract Algebra, Vol. III, D. Van Nostrand, New York, 1964.
- I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
- M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.
