Almost ϕ-integrally closed rings

Abstract

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

Keywords:

• Almost integrally closed domain
• almost $\varphi$-integrally closed ring
• $\varphi$-integrally closed ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13B02
• 13B22
• Secondary: 13B30
• 13A15

Additional information

Funding

Author Rahul Kumar was supported by the Research Initiation Grant Scheme from Birla Institute of Technology and Science Pilani, Pilani, India. Author Anant Singh was supported by a Grant from CSIR India, No. 09/0045(12157)/2021-EMR-I. Author Atul Gaur was supported by a grant (Ref. No./IoE/2021/12/FRP) from University of Delhi.