Notes on PrÜfer v-Multiplication Rings

Let D be an integral domain (∋1) and K be the quotient field of D. Let F(D) be the set of nonzero fractional ideals of D. A mapping * of F(D) into itself is called *-operation on D if it satisfies: (1) for 0_??_a∈K and a∈F(D), we have (a)* =(a) and (aa)*=aa*; (2) for a∈F(D), we have a⊂a*(⊂ means ⊆) and, a⊂b implies a*⊂b*; (3) for a∈F(D), we have (a*)*=a*. For example, a mapping a_??_(a-1)-1 is a *-operation, and is called v-operation. For a subset b⊂K, we denote {x∈K; xb⊂A} by b-1 in general. Let * be a *-operation on D. The mapping (a, b)_??_(ab)* is called *-product. If{a*; 0_??_a finitely generated} makes a group under the *-product, D is called prufer *-multiplication domain. As to the conditions and the related properties under which D becomes a prufer *-multi-