INC-EXTENSIONS AMID ZERO-DIVISORS

Abstract

All rings considered are commutative with identity. Let R be a complemented ring with integral closure R0 (in its total quotient ring K). Then R ⊆ S satisfies INC for each overring S of R (inside K) if and only if R0 is a Prüfer ring. If R0 is a Prüfer ring and T is a complemented ring that contains R as a subring such that each regular element of T is a root of a polynomial in R[X] with a regular coefficient and T is torsion-free over R, then R ⊆ T satisfies INC. As a consequence, a new generalization for rings with nontrivial zero-divisors is found of Pr¨ufer’s result on the integral closure of a Prüfer domain in a field extension of the quotient field.