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\(UN\)-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.

MSC:

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)

Citations:

Zbl 1156.16303
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References:

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