# 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