ABSTRACT
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 ϕ chained ring (ϕ-CR) if for every x,y ∈ R\Nil(R), either x | y or y | x. A prime ideal P of R is said to be a ϕ-strongly prime ideal if for every a,b ∈ R\Nil(R), either a|b or aP ⊂ bP. In this paper, we show that if R admits a regular ϕ-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 ϕ-CR or R admits a minimal regular prime ideal Q and c(R) = (Q : Q) is a ϕ-CR with maximal ideal Q. We also prove that the complete integral closure of a conducive domain is a valuation domain.
