The Content of a Gaussian Polynomial Is Invertible

Let R be an integral domain and let f(X) be a nonzero polynomial in R[X ]. The content of f is the ideal c(f) generated by the coefficients of f . The polynomial f(X) is called Gaussian if c(fg) = c(f)c(g) for all g(X) ∈ R[X ]. It is well known that if c(f) is an invertible ideal, then f is Gaussian. In this note we prove the converse. Let R be a ring, that is, a commutative ring with unity. Let A be a ring extension of R, and let f(X) ∈ A[X] be a polynomial: f(X) = anX n + . . .+ a1X + a0. The content ideal of f , designated by c(f) = cR(f), is the R-submodule of A generated by the coefficients of f in A. It is easy to see that if f, g ∈ R[X] (and A = R), then c(fg) ⊆ c(f)c(g). The inclusion in this statement is generally proper. The polynomial f(X) ∈ R[X] is said to be Gaussian if c(fg) = c(f)c(g) holds for all g(X) ∈ R[X]. It is well known that if c(f) is an invertible ideal of R, then f is Gaussian. More generally, if R is any ring and c(f) is locally principal, then f is Gaussian (see, e.g., [1, Theorem 1.1]). Recall that a nonzero ideal I of an integral domain R is invertible iff it is locally principal, that is, iff IRM is a principal ideal for each maximal ideal M of R. For general background see [3]. It has been conjectured that the converse is true if R is an integral domain (see [1, 4]), that is, a Gaussian polynomial over an integral domain has an invertible content. This question is included in the Ph.D. thesis of Kaplansky’s student H. T. Tang. Significant progress has been made on this conjecture in two recent papers [4, 5]: in [4], Glaz and Vasconcelos prove the conjecture for R integrally closed with some additional assumptions (including the Noetherian case). The general Noetherian case is settled in [5]. As explained below, the conjecture 2000 Mathematics Subject Classification. 13 B25.