Rings in which all elements are the sum of a central element and an element from $Δ(R)$
Peter Danchev, Arash Javan, Omid Hasanzadeh, Ahmad Moussavi
TL;DR
The paper introduces and develops the theory of $CΔ$ rings, where every element decomposes as a central part plus a $Δ(R)$-part, extending prior notions such as $CJ$, $CU$, and $CN$. It establishes core structural properties, including quotient behavior, nilpotent and radical inclusions, and liftings to corner subrings, and proves that exchange $CΔ$ rings are clean while $R[[x]]$ is $CΔ$ iff $R$ is. It then analyzes a wide array of extensions—matrix-type rings, trivial and skew triangular rings, and related constructions—providing criteria for when these inherit the $CΔ$ property and presenting several counterexamples. The results tie the $CΔ$ property to the Köthe conjecture and to decompositions in related ring classes, offering new tools for ring-structure analysis and several open questions for future work.
Abstract
We define and consider in-depth the so-called $CΔ$ rings as those rings $R$ whose elements are a sum of an element in $C(R)$ and of an element in $Δ(R)$. Our achieved results somewhat strengthen these recently obtained by Ma-Wang-Leroy in Czechoslovak Math. J. (2024) as well as these due to Kurtulmaz-Halicioglu-Harmanci-Chen in Bull. Belg. Math. Soc. Simon Stevin (2019). Specifically, we succeeded to establish that exchange $CΔ$ 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 $CΔ$ if, and only if, so is $R$. And, furthermore, we show that, for any ring $R$, if the polynomial ring $R[x]$ is a $CΔ$ ring, then $R$ satisfies the Köthe conjecture. Some other closely related things concerning certain extensions of $CΔ$ rings are also presented.