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<p>It is conjectured that, for any group <i>G</i>, the Jacobson radical <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(J({\mathbb {Z}}[G])\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">[</mo> <mi>G</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the integral group ring <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {Z}}[G]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">[</mo> <mi>G</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> is zero. This is known to be true when <i>G</i> is finite. Here we show it is true for a reasonably large class of infinite groups, including finitely generated linear groups and groups which satisfy Higman’s ‘two unique products’ condition.</p>

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On the radical of group rings

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月7日

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  • F. E. A. Johnson

摘要

It is conjectured that, for any group G, the Jacobson radical \(J({\mathbb {Z}}[G])\) J ( Z [ G ] ) of the integral group ring \({\mathbb {Z}}[G]\) Z [ G ] is zero. This is known to be true when G is finite. Here we show it is true for a reasonably large class of infinite groups, including finitely generated linear groups and groups which satisfy Higman’s ‘two unique products’ condition.