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Ring in which every element is sum of two $6-$potent elements | ||
| Journal of Algebra and Related Topics | ||
| دوره 14، شماره 1، آبان 2026، صفحه 77-86 اصل مقاله (157.08 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2024.26096.1605 | ||
| نویسندگان | ||
| K. N. Deka* ؛ H. K. Saikia | ||
| Department of Mathematics, Gauhati University, Guwahati, India | ||
| چکیده | ||
| In this paper, we prove the following results. Every element of a ring $R$ is a sum of two commuting $6-$potent elements if and only if $R$ is isomorphic to $R_1\times R_2\times R_3$, where $R_1$ is isomorphic to a subdirect product of $Z_2$'s, $R_2$ is isomorphic to a subdirect product of $Z_3$'s and $R_3$ is isomorphic to a subdirect product of $Z_{11}$'s. Also, if every element of a ring $R$ is the sum of two 6-potent and one nilpotent all commute with each other, then $R$ is isomorphic to $R_1\times R_2\times R_3$, where $J(R_1)$ is nil and $R_1/J(R_1)$ is a subdirect product of rings isomorphic to either of the rings $Z_2,F_4,M_2(F_2)$ and $M_2(F_4)$ , $a^{81}-a$ is nilpotent for every $a\in R_2$ , $J(R_3)$ is nil and $R_3/J(R_3)$ is a subdirect product of $Z_{11}$'s. | ||
| کلیدواژهها | ||
| 4-potents؛ 6-potents؛ Chinese Remainder Theorem | ||
| مراجع | ||
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[1] A. Diesl, Sums of commuting potent and nilpotent elements in rings, Journal of Algebra and Its Applications, (5) 22 (2023). [2] M. T. Koşan, T. Yildirim and Y. Zhou, Rings with xn −x nilpotent, Journal of Algebra and Its Applications, (4) 19 (2020). [3] Y. Hirano and H. Tominaga, Rings in which every element is the sum of two idempotents, Bull. Austral. Math. Soc., (2) 37 (1988), 161–164. [4] Z. Ying, T. Kosan and Y. Zhou, Rings in which every element is a sum of two tripotents, Canad. Math. Bull., (3) 59 (2016), 661-672. | ||
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