Exceptional and complementable units in rings

Abstract A unit u in a ring R is called exceptional if $$1-u$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> is also a unit. Such units have been previously studied from a Number Theory perspective. In this paper, we investigate exceptional units from the standpoint of Ring Theory, with particular emphasis on matrix rings, where several characterizations are provided. We define and explore a special subclass of exceptional units, termed complementable units. An exceptional unit u is called complementable if $$u(1-u)=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We determine the residue class rings that contain complementable units and identify such units in certain matrix rings.