\(UN\)-rings. (English) Zbl 1397.16037

Summary: A nonzero ring is called a \(UN\)-ring if every nonunit is a product of a unit and a nilpotent element. We show that all simple Artinian rings are \(UN\)-rings and that the \(UN\)-rings whose identity is a sum of two units (e.g. if 2 is a unit), form a proper class of 2-good rings (in the sense of P. Vámos [Q. J. Math. 56, No. 3, 417–430 (2005; Zbl 1156.16303)]). Thus, any noninvertible matrix over a division ring is the product of an invertible matrix and a nilpotent matrix.


16U99 Conditions on elements
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
15B33 Matrices over special rings (quaternions, finite fields, etc.)
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16U60 Units, groups of units (associative rings and algebras)


Zbl 1156.16303
Full Text: DOI


[1] 1. G. Călugăreanu and T. Y. Lam, Fine rings: A new class of simple rings, J. Algebra and Its Appp.15(9) (2015) (18 pages). genRefLink(128, ’S0219498816501826BIB001’, ’000383799800015’);
[2] 2. G. Ehrlich, Unit-regular rings, Portugal. Math.27 (1968) 209-212. · Zbl 0201.03901
[3] 3. M. Henriksen, Two classes of rings generated by their units, J. Algebra31 (1974) 182-193. genRefLink(16, ’S0219498816501826BIB003’, ’10.1016
[4] 4. I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc.66 (1949) 464-491. genRefLink(16, ’S0219498816501826BIB004’, ’10.1090
[5] 5. T. Y. Lam, A First Course in Noncommutative Ring Theory, 2nd edn. Graduate Texts in Mathematics, Vol. 131 (Springer-Verlag, Berlin, 2001). genRefLink(16, ’S0219498816501826BIB005’, ’10.1007 · Zbl 0980.16001
[6] 6. R. Raphael, Rings which are generated by units, J. Algebra28 (1974) 199-205. genRefLink(16, ’S0219498816501826BIB006’, ’10.1016
[7] 7. P. Vámos, 2-Good rings, Quart. J. Math.56 (2005) 417-430. genRefLink(16, ’S0219498816501826BIB007’, ’10.1093
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