Prüfer ∗-multiplication domains, Nagata rings, and Kronecker function rings

Let D be an integrally closed domain, ∗ a star-operation on D, X an indeterminate over D, and N∗={f∈D[X]|∗(Af)=D}. For an e.a.b. star-operation ∗1 on D, let Kr(D,∗1) be the Kronecker function ring of D with respect to ∗1. In this paper, we use ∗ to define a new e.a.b. star-operation ∗c on D. Then we prove that D is a Prüfer ∗-multiplication domain if and only if D[X]N∗=Kr(D,∗c), if and only if Kr(D,∗c) is a quotient ring of D[X], if and only if Kr(D,∗c) is a flat D[X]-module, if and only if each ∗-linked overring of D is a Prüfer v-multiplication domain. This is a generalization of the following well-known fact that if D is a v-domain, then D is a Prüfer v-multiplication domain if and only if Kr(D,v)=D[X]Nv, if and only if Kr(D,v) is a quotient ring of D[X], if and only if Kr(D,v) is a flat D[X]-module.