A question about maximal non ϕ -chained subrings

Abstract

Let H0 be the set of rings R such that Nil(R)=Z(R) is a divided prime ideal of R. The concept of maximal non ϕ-chained subrings is a generalization of maximal non valuation subrings from domains to rings in H0. This generalization was introduced in \cite{rahul} where the authors proved that if R∈H0 is an integrally closed ring with finite Krull dimension, then R is a maximal non ϕ-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 ϕ-chained subring of some overring S. The existence of such an overring S of R is shown for 3≤n≤6, and no such overring exists for n=7.