Extensions of integral domains and quasi-valuations
Let S be an integral domain with field of fractions F, and let A be an F-algebra having an S-stable basis. We prove the existence of an S-subalgebra R of A lying over S whose localization with respect to S is A (we call such R an S-nice subalgebra of A). We also show that there is no such minimal S-nice subalgebra of A. Given a valuation v on F with a corresponding valuation domain Ov, and an Ov-stable basis of A over F, we prove the existence of a quasi-valuation on A extending v on F. Moreover, we prove the existence of an infinite decreasing chain of quasi-valuations on A, all of which extend v. Finally, we present applications for the above existence theorems; for example, we show that if A is commutative and C is any chain of prime ideals of S, then there exists an S-nice subalgebra of A having a chain of prime ideals covering C.