Flatness, LCM-stability, and related module-theoretic properties

AbstractThe relationship between flatness and LCM-stability is clarified by the following two results. A finitely generated ideal I of an integral domain is flat if and only if I is n-flat for some integer n ⩾ 2. There exists an integral domain with a nonflat ideal J = (a, b, c) such that Jab ∩ Jac ∩ Jbc = J(ab, ac, bc). Next, related module-theoretic properties, (∗∗) and (∗), respectively weaker than projectivity and flatness, are introduced. Under appropriate finiteness conditions, these properties are preserved by certain inverse limits. Their study leads to new characterizations of quasicomplete local rings and coherent integral domains