Rings in which all elements are the sum of a central element and an element from (R)
Abstract
We define and consider in-depth the so-called C rings as those rings R whose elements are a sum of an element in C(R) and of an element in (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 C 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 C if, and only if, so is R. And, furthermore, we show that, for any ring R, if the polynomial ring R[x] is a C ring, then R satisfies the K\"othe conjecture. Some other closely related things concerning certain extensions of C rings are also presented.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.