# Extensions of integral domains and quasi-valuations

Seminar
Speaker
Dr. Shai Sarussi (Sami Shamoon College of Engineering)
Date
07/11/2018 - 11:30 - 10:30Add to Calendar 2018-11-07 10:30:00 2018-11-07 11:30:00 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. Third floor seminar room אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Third floor seminar room
Abstract

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.

תאריך עדכון אחרון : 30/10/2018