Kronecker function rings and Prüfer extension

Let R be a subring of S. The classical Kronecker function ring when R is a domain and S its quotient field associates to the domain R a Bezout domain (that is, a domain in which every finitely generated ideal is principal). Knebusch and Kaiser introduce a notion of star operation ★ on the commutative ring extension R ⊆ S and generalize the concept of Kronecker function ring K_r1(★) from integral domains to commutative ring extensions. We introduce a new ring K_r2(★) which is a subring of K_r1(★) and we will show with an example that K_r1(★) ≠ K_r(★). We also discuss the properties of ring extensions using the star operation ★ and in particular, we focus on the case where R ⊆ S is a Prüfer extension. Joint work with Simplice Tchamna (Georgia College & State University).