Title

Note on Cyclotomic Polynomials and Prime Ideals

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ℤ-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ⊂ Q of A such that ht(Q/P) = 2.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1081/agb-120027870

Keywords

commutative ring, cyclotomic polynomials, euler phi-function, homogeneous prime ideal, noetherian ring, radical ideal, resultant

Publication Date

2004

Recommended Citation

Ratliff Jr, Louis J., David E. Rush, and Kishor Shah. "Note on cyclotomic polynomials and prime ideals." Communications in Algebra 32, no. 1 (2004): 333-343.

Journal Title

Communications in Algebra

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