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<p>An element is called having a <i>CQ</i>-decomposition if it is the sum of a central element and a quasi-nilpotent element. A ring <i>R</i> is called a <i>CQ</i> ring if each element of <i>R</i> has a <i>CQ</i>-decomposition. We establish the basic properties of <i>CQ</i> rings, and give several characterizations of these rings. It is shown that every <i>CQ</i> ring is Dedekind finite. Moreover, we also prove that <i>R</i> is a <i>CQ</i> ring iff <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(U(R)=[U(R)\cap C(R)]+Q(R)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and each element is the sum of a central element and a unit element.</p>

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Rings in which all elements are the sum of a central element and a quasi-nilpotent element

  • 期刊论文

  • 发表日期：

    2025年5月19日

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  • Phan Hong Tin,
  • Nguyen Quoc Tien

摘要

An element is called having a CQ-decomposition if it is the sum of a central element and a quasi-nilpotent element. A ring R is called a CQ ring if each element of R has a CQ-decomposition. We establish the basic properties of CQ rings, and give several characterizations of these rings. It is shown that every CQ ring is Dedekind finite. Moreover, we also prove that R is a CQ ring iff \(U(R)=[U(R)\cap C(R)]+Q(R)\) U ( R ) = [ U ( R ) ∩ C ( R ) ] + Q ( R ) and each element is the sum of a central element and a unit element.