Geodesic
Nil-clean and unit-regular elements in certain subrings of ${\mathbb M}_2(\mathbb Z)$
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 197-205 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson's lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl's question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are not clean in a ring. The rings under consideration in our examples are particular subrings of $\mathbb {M}_2(\mathbb {Z})$.
An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson's lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl's question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are not clean in a ring. The rings under consideration in our examples are particular subrings of $\mathbb {M}_2(\mathbb {Z})$.
DOI : 10.21136/CMJ.2018.0256-17
Classification : 11D09, 16S50, 16U60
Keywords: clean element; nil-clean element; unit-regular element; Jacobson's lemma for nil-clean elements 
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Wu, Yansheng; Tang, Gaohua; Deng, Guixin; Zhou, Yiqiang. Nil-clean and unit-regular elements in certain subrings of ${\mathbb M}_2(\mathbb Z)$. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 197-205. doi: 10.21136/CMJ.2018.0256-17

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