Geodesic
Rings in which elements are sum of a central element and an element in the Jacobson radical
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 515-533
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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.
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.
DOI : 10.21136/CMJ.2024.0433-23
Classification : 16N20, 16N40, 16U70
Keywords: CJ ring; center; Jacobson radical; clean ring 
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     title = {Rings in which elements are sum of a central element and an element in the {Jacobson} radical},
     journal = {Czechoslovak Mathematical Journal},
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     year = {2024},
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Ma, Guanglin; Wang, Yao; Leroy, André. Rings in which elements are sum of a central element and an element in the Jacobson radical. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 515-533. doi: 10.21136/CMJ.2024.0433-23

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