REMARKS ON A GOLDBACH PROPERTY

Abstract

In this paper, we study Noetherian Boolean rings. We show that if R is a Noetherian Boolean ring, then R is finite and R ≃ (Z2)n for some integer n ≥ 1. If R is a Noetherian ring, then R/J is a Noetherian Boolean ring, where J is the intersection of all ideals I of R with |R/I| = 2. Thus R/J is finite, and hence the set of ideals I of R with |R/I| = 2 is finite. We also give a short proof of Hayes’s result : For every polynomial f(x) ∈ Z[x] of degree n ≥ 1, there are irreducible polynomials g(x) and h(x), each of degree n, such that g(x) + h(x) = f (x).