Note on Cyclotomic Polynomials and Prime Ideals

Louis J. Ratliff
David E. Rush
Kishor Shah, Missouri State University

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.