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)→𝑅Nil⁡(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.