ON POLYNOMIAL RINGS WITH A GOLDBACH PROPERTY

Abstract. David Hayes observed in 1965 that when R = Z, every element of R[T] of degree n ≥ 1 is a sum of two irreducibles in R[T] of degree n. We show that this result continues to hold for any Noetherian domain R with infinitely many maximal ideals. It appears that David Hayes [5] was the first to observe the following polynomial analogue of the celebrated Goldbach conjecture: If R = Z, then every element of R[T] of degree n ≥ 1 can be written as the sum of two irreducibles of degree n. His proof is a clever application of Eisenstein’s irreducibility criterion. Hayes’s theorem and its proof were rediscovered by Rattan and Stewart [10] (see also [1] for some cognate results). Recently Saidak [11] and Kozek [7] have considered quantitative variants of Hayes’s theorem. The latter shows that in a precise asymptotic sense, for a given monic polynomial A(T) ∈ Z[T] of degree n ≥ 2, almost all (100%) of its representations as a sum of two monic polynomials are such that both summands are irreducible. In this note we consider a generalization in a different direction. Namely, we investigate which integral domains R have the property (?). In [5], Hayes points out that his proo