Generalizing Semi-$n$-Potent Rings

We define and explore the class of rings $R$ for which each element in $R$ is a sum of a tripotent element from $R$ and an element from the subring $\mathrm{\Delta }(R)$ of $R$ which commute each other. Succeeding to obtain a complete description of these rings modulo their Jacobson radical as the direct product of a Boolean ring and a Yaqub ring, our results somewhat generalize those established by Koşan-Yildirim-Zhou in Can. Math. Bull. (2019).