Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, first online, pp. 1-19

Rings in which elements are sum of a central element and an element in the Jacobson radical

Guanglin Ma, Yao Wang, André Leroy

Received September 21, 2023. Published online February 13, 2024.

Abstract: An element in a ring $R$ is called CJ if it is of the form $c+j$, where $c$ belongs to the center and $j$ is an element from the Jacobson radical. A ring $R$ is called CJ if each element of $R$ is CJ. We establish the basic properties of CJ rings, give several characterizations of these rings, and connect this notion with many standard elementwise properties such as clean, uniquely clean, nil clean, CN, and CU. We study the behavior of this notion under various ring extensions. In particular, we show that the subring $C+J$ is always a CJ ring and that if $R[x]$ is a CJ ring then $R$ satisfies the Köthe conjecture.

Keywords: CJ ring; center; Jacobson radical; clean ring

Classification MSC: 16U70, 16N20, 16N40

DOI: 10.21136/CMJ.2024.0433-23

PDF available at: Springer Institute of Mathematics CAS

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Affiliations: Guanglin Ma (corresponding author), Yao Wang, School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Shangxian Building No.725, Nanjing, P. R. China, e-mail: linlinguangma@163.com, wangyao@nuist.edu.cn; André Leroy, Laboratoire Mathématiques Lens (LML), University of Artois, U. R. 2462, Rue Jean Souvraz SP18, 62307 Lens, France, e-mail: andre.leroy@univ-artois.fr

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