A corrigendum to “Hereditary properties between a ring and its maximal subrings”

Abstract

Let $R$ be a commutative ring with identity. In [2] (Proposition 3.1), Azarang proved that if $R$ is an integral domain and $S$ is a maximal subring of $R$, and is integrally closed in $R$, then $\mathrm{d}\mathrm{i}\mathrm{m}(S) = 1$ implies that $\mathrm{d}\mathrm{i}\mathrm{m}(R) = 1$ if and only if $(S : R) = 0$. An example is given which shows the above mentioned proposition is not correct.