Construction of Ideal Systems with Nice Noetherian Properties

Abstract

Recently, W. Fanggui and R. L. McCasland [F-McC1], [F-McC2] introduced the notion of a w-envelope M _w of a non-zero torsion-free module M over an integral domain D as follows: If K is a quotient field of D and V = KM is the vector space generated by M, then M _w consists of all x ∈ V such that Jx ⊂ M for some finitely generated ideal J⊲D satisfying J ^-1 = D. They called an ideal I ⊲ D a w-ideal if I _w = I, and they called D a strong Mori domain if D satisfies the ACC on w-ideals. As a main result, they proved that the w-ideals of a strong Mori domain have a primary decomposition and satisfy the Krull Intersection Theorem and the Krull Principal Ideal Theorem.