Abstract
We define and consider in-depth the so-called “$C\Delta$ rings” as those rings $R$ whose elements are a sum of an element in $C(R)$ and of an element in $\Delta(R)$. Our results somewhat strengthen those recently obtained by Ma – Wang – Leroy in 2024 as well as those due to Kurtulmaz – Halicioglu – Harmanci – Chen in 2019. Specifically, we succeeded to establish that exchange $C\Delta$ rings are always clean as well as that exchange CN rings are strongly clean. Likewise, we prove that, for any ring $R$, the ring of formal power series $R[[x]]$ over $R$ is if and only if $R$ is also $C\Delta$. And, furthermore, we show that, for any ring $R$, if the polynomial ring $R[x]$ is a $C\Delta$ ring, then $R$ satisfies the Köthe conjecture. Some other closely related phenomena concerning certain extensions of $C\Delta$ rings are also presented.