The A + XB[X] construction and G-Dedekind domains

Let [Formula: see text] be an extension of domains, [Formula: see text] be an indeterminate over [Formula: see text]. Let [Formula: see text] be a fixed integer such that [Formula: see text] and [Formula: see text]. We prove that [Formula: see text] is a [Formula: see text]-Dedekind domain (NWF domain) if and only if both [Formula: see text] and [Formula: see text] are fields and [Formula: see text]. We also show that if [Formula: see text] is a [Formula: see text]-Dedekind domain, then every projective ideal of [Formula: see text] is principal.