The FIP Property for Commutative Rings

Dr. Bernadette Mullins

Department of Mathematics
Birmingham Southern College

December 5, 2003
202 Madison Hall
3:00 PM (Coffee and Cookies at 2:30)

Abstract

Let R T be an extension of commutative rings. We say that R T has FIP (for the "finite intermediate property") if the set of R-subalgebras of T is finite. This study arises as a generalization of the classical Primitive Element Theorem. If R T has FIP and T is an integral domain, then either R and T are fields or T is the quotient field of R. If R is a field, then the main result identifies four exhaustive cases which serve to characterize the condition that R T has FIP. Some results are also available in case R is neither a field nor a domain.