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<p>An element <i>a</i> of a ring <i>R</i> is said to have <i>left central stable range one</i> if whenever <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Ra+Rb=R\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>a</mi> <mo>+</mo> <mi>R</mi> <mi>b</mi> <mo>=</mo> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(b\in R\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>∈</mo> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(z\in Z(R)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo>∈</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (the center of <i>R</i>) such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(a+zb\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>+</mo> <mi>z</mi> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation> is a unit of <i>R</i>. A ring <i>R</i> has <i>left central stable range one</i> if all its elements have it. In this work, elements and rings with left central stable range one are studied. The condition is left-right symmetric for such rings.</p>

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Central stable range one for elements and rings

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  • 发表日期：

    2025年6月30日

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  • Grigore Călugăreanu

摘要

An element a of a ring R is said to have left central stable range one if whenever \(Ra+Rb=R\) R a + R b = R for any \(b\in R\) b ∈ R , there exists \(z\in Z(R)\) z ∈ Z ( R ) (the center of R) such that \(a+zb\) a + z b is a unit of R. A ring R has left central stable range one if all its elements have it. In this work, elements and rings with left central stable range one are studied. The condition is left-right symmetric for such rings.