On the FIP Property for Extensions of Commutative Rings #

Abstract

ABSTRACT A (unital) extension R ⊆ T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ⊂ S ⊂ T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ⊂ T is a (module-) finite minimal ring extension, then R(X)⊂T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ⊆ T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ⊆ R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ⊆ R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ℤ which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (ℤ:R)≠0. Some results are also given for the residually FIP property.