A Schreier Domain Type Condition II

Abstract

Let D be an integral domain and ∗ a star operation on D. We study the domains characterized by the following property: whenever I ⊇ AB with I, A, B nonzero ideals, there exist nonzero ideals H and J such that I∗ = (HJ)∗, H∗ ⊇ A and J∗ ⊇ B. We call them ∗-sharp domains. We show that D is t-sharp if and only if D[X] is t-sharp if and only if D[X]Nv is d-sharp, where Nv is the multiplicative set of D[X] consisting of all nonzero polynomials a0 + a1X + ⋯ + anXn such that (a0,a1,…,an)v = D.