Bezout Overrings of a Polynomial Ring D[{Xα}

Abstract

Let D be an integral domain with quotient field K, Λ be an index set with |Λ| ≥1, {Xα∣α ∈ Λ} be a set of indeterminates over D, {Vλ} be the set of valuation overrings of D, andIt is known that R2 is a Bezout domain. In this article, we show that R1 is a PID and R is a Bezout domain. We also show that D is a Prüfer domain if and only if R = D[{Xα}]S, if and only if D[{Xα}]S is a Bezout domain, where .