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A nil-clean \(2\times 2\) matrix over the integers which is not clean. (English) Zbl 1294.16019

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.

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)
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[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)
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