On the complete integral closure of rings that admit φ-strongly prime ideal

Let R be a commutative ring with 1 and T(R) be its total quotient ring such that Nil(R) (the set of all nilpotent elements of R) is a divided prime ideal of R. Then R is called a phi-chained ring (phi-CR) if for every x, y is an element of R \ Nil (R), either x \ y or y \ x. A prime ideal P of R is said to be a phi-strongly prime ideal if for every a, b is an element of R \ Nil(R), either a \ b or aP subset of bP. In this paper, we show that if R admits a regular phi-strongly prime ideal, then either R does not admit a minimal regular prime ideal and c(R) (the complete integral closure of R inside T(R)) = T(R,) is a phi-CR or R admits a minimal regular prime ideal Q and c(R) = (Q : Q) is a phi-CR with maximal ideal Q. We also prove that the complete integral closure of a conducive domain is a valuation domain.