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<p>A unit <i>u</i> in a ring <i>R</i> is called exceptional if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1-u\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>-</mo> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> 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 <i>u</i> is called complementable if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(u(1-u)=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We determine the residue class rings that contain complementable units and identify such units in certain matrix rings.</p>

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Exceptional and complementable units in rings

  • 期刊论文

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  • 发表日期：

    2026年5月19日

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  • Grigore Călugăreanu

摘要

A unit u in a ring R is called exceptional if \(1-u\) 1 - u 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 \(u(1-u)=1\) u ( 1 - u ) = 1 . We determine the residue class rings that contain complementable units and identify such units in certain matrix rings.