On purely-maximal ideals with applications

Let $B$ be a ring of the form $B=A+J$ where $A$ is a subring of $B, J$ is an ideal of $B$ such that $J\cap A=0$ and $1+J\subseteq\mathcal{U}(B)$ the set of units of $B$. Let $C$ be a subring of $B$ containing $A$. We prove that purely-maximal ideals of $C$ are exactly $I C$ where $I$ ranges over purely-maximal ideals of $A$. We deduce that $C$ is semi-Noetherian if and only if $A$ is semi-Noetherian. We show that Tarizadeh and Aghajani's conjecture holds in $C$ if and only if it holds in $A$. As an application, we generalize all results in [N. Ouni and A. Benhissi, Beitr. Algebra Geom., 65(1)(2024), 229-240] and we study purely-maximal ideals of an amalgamation ring along an ideal.