Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Danchev P., Javan A., Hasanzadeh O., Moussavi A. Rings R in which all elements are the sum of a central element and an element from Δ(R). Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2026, vol. 26, iss. 3, pp. 335-349. DOI: 10.18500/1816-9791-2026-26-3-335-349, EDN: IQDWRM

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2026
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Language: 
English
Heading: 
Mathematics
Article type: 
Article
UDC: 
512.71
DOI: 
10.18500/1816-9791-2026-26-3-335-349
EDN: 
IQDWRM

Rings R in which all elements are the sum of a central element and an element from Δ(R)

Autors: 
Danchev Peter, Bulgarian Academy of Sciences
Javan Arash, Department of Mathematics, Tarbiat Modares University, Jalal Al-Ahmad Nasr
Hasanzadeh Omid, Department of Mathematics, Tarbiat Modares University, Jalal Al-Ahmad Nasr
Moussavi Ahmad, Department of Mathematics, Tarbiat Modares University, Jalal Al-Ahmad Nasr
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.

Key words: 
central element
Δ(R) subring
CΔ ring
matrix ring
Acknowledgments: 
This work of the first-named author, Peter Danchev, is partially supported by the Junta de Andalucía (project No. FQM 264). This work is based upon research funded by Iran National Science Foundation (INSF) (project No. 40402401).
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Received: 
09.08.2025
Accepted: 
16.11.2025
Published: 
31.08.2026
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