Andrica, Dorin; Călugăreanu, Grigore A nil-clean \(2\times 2\) matrix over the integers which is not clean. (English) Zbl 1294.16019 J. Algebra Appl. 13, No. 6, Article ID 1450009, 9 p. (2014). Summary: While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring \(\mathcal M_2(\mathbb Z)\) which is not clean. Cited in 2 ReviewsCited in 8 Documents MSC: 16S50 Endomorphism rings; matrix rings 16N40 Nil and nilpotent radicals, sets, ideals, associative rings 16U60 Units, groups of units (associative rings and algebras) 16U80 Generalizations of commutativity (associative rings and algebras) Keywords:matrix rings; nil clean rings; units; idempotents; nilpotent elements; nil clean elements; nil clean matrices; Diophantine equations; Pell equation; unit regular rings PDFBibTeX XMLCite \textit{D. Andrica} and \textit{G. Călugăreanu}, J. Algebra Appl. 13, No. 6, Article ID 1450009, 9 p. (2014; Zbl 1294.16019) Full Text: DOI References: [1] DOI: 10.1007/978-0-8176-4549-6 · Zbl 1226.11001 · doi:10.1007/978-0-8176-4549-6 [2] DOI: 10.1081/AGB-100002185 · Zbl 0992.16011 · doi:10.1081/AGB-100002185 [3] DOI: 10.1016/j.jalgebra.2013.02.020 · Zbl 1296.16016 · doi:10.1016/j.jalgebra.2013.02.020 [4] DOI: 10.1016/j.jalgebra.2004.04.019 · Zbl 1067.16050 · doi:10.1016/j.jalgebra.2004.04.019 [5] Nagell I., Introduction to Number Theory (1951) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.