Communications of the
Korean Mathematical Society CKMS

Let $\mathcal{H}_0$ be the set of rings $R$ such that $Nil(R) = Z(R)$ is a divided prime ideal of $R$. The concept of maximal non $\phi$-chained subrings is a generalization of maximal non valuation subrings from domains to rings in $\mathcal{H}_0$. This generalization was introduced in \cite{rahul} where the authors proved that if $R \in \mathcal{H}_0$ is an integrally closed ring with finite Krull dimension, then $R$ is a maximal non $\phi$-chained subring of $T(R)$ if and only if $R$ is not local and $|[R, T(R)]|$ = $\dim (R) + 3$. This motivates us to investigate the other natural numbers $n$ for which $R$ is a maximal non $\phi$-chained subring of some overring $S$. The existence of such an overring $S$ of $R$ is shown for $3\leq n \leq 6$, and no such overring exists for $n = 7$.

Keywords: Maximal non $\phi$-chained ring, integrally closed ring, $\phi$-Pr\"ufer ring

MSC numbers: Primary 13B02, 13B22

Supported by: The first author was supported by the MATRICS grant from DST-SERB, No. MTR/2018 /000707 and the second author was supported by the Research Initiation Grant Scheme from Birla Institute of Technology and Science Pilani, Pilani, India.