Geodesic
Note on strongly nil clean elements in rings
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 87-92 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $R$ be an associative unital ring and let $a\in R$ be a strongly nil clean element. We introduce a new idea for examining the properties of these elements. This approach allows us to generalize some results on nil clean and strongly nil clean rings. Also, using this technique many previous proofs can be significantly shortened. Some shorter proofs concerning nil clean elements in rings in general, and in matrix rings in particular, are presented, together with some generalizations of these results.
Let $R$ be an associative unital ring and let $a\in R$ be a strongly nil clean element. We introduce a new idea for examining the properties of these elements. This approach allows us to generalize some results on nil clean and strongly nil clean rings. Also, using this technique many previous proofs can be significantly shortened. Some shorter proofs concerning nil clean elements in rings in general, and in matrix rings in particular, are presented, together with some generalizations of these results.
DOI : 10.21136/CMJ.2018.0167-17
Classification : 13B25, 15B33, 16U99
Keywords: nilpotent element; nil clean element 
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Kostić, Aleksandra; Petrović, Zoran Z.; Pucanović, Zoran S.; Roslavcev, Maja. Note on strongly nil clean elements in rings. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 87-92. doi: 10.21136/CMJ.2018.0167-17

[1] Chen, H.: Some strongly nil clean matrices over local rings. Bull. Korean Math. Soc. 48 (2011), 759-767. | DOI | MR | JFM

[2] Chen, H.: Strongly nil clean matrices over $R[x]/(x^2-1)$. Bull. Korean Math. Soc. 49 (2012), 589-599. | DOI | MR | JFM

[3] Chen, H.: On strongly nil clean matrices. Commun. Algebra 41 (2013), 1074-1086. | DOI | MR | JFM

[4] Chen, H., Sheibani, M.: On strongly nil clean rings. Commun. Algebra 45 (2017), 1719-1726. | DOI | MR | JFM

[5] Diesl, A. J.: Nil clean rings. J. Algebra 383 (2013), 197-211. | DOI | MR | JFM

[6] Hirano, Y., Tominaga, H., Yaqub, A.: On rings in which every element is uniquely expressible as a sum of a nilpotent element and a certain potent element. Math. J. Okayama Univ. 30 (1988), 33-40. | MR | JFM

[7] Koşan, T., Wang, Z., Zhou, Y.: Nil-clean and strongly nil-clean rings. J. Pure Appl. Algebra 220 (2016), 633-646. | DOI | MR | JFM

[8] Šter, J.: Rings in which nilpotents form a subring. Carpathian J. Math. 32 (2016), 251-258. | MR

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