Almost ϕ-integrally closed rings

Abstract

Let R be a commutative ring with unity. The notion of almost ϕ -integrally closed ring is introduced which generalizes the concept of almost integrally closed domain. Let H be the set of all rings such that Nil ( R ) is a divided prime ideal of R and ϕ : T ( R ) → R Nil ( R ) is a ring homomorphism defined as ϕ ( x ) = x for all x ∈ T ( R ) . A ring R ∈ H is said to be an almost ϕ -integrally closed ring if ϕ ( R ) is integrally closed in ϕ ( R ) ϕ ( p ) for each nonnil prime ideal p of R. Using the idealization theory of Nagata, examples are also given to strengthen the concept.