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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((R, \mathfrak {m})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi mathvariant="fraktur">m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be an one-dimensional local Cohen–Macaulay ring. An <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathfrak m\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">m</mi> </math></EquationSource> </InlineEquation>-primary ideal <i>I</i> of <i>R</i> is Elias if the types of <i>I</i> and of <i>R</i>/<i>I</i> are equal. Canonical and principal ideals are Elias, and Elias ideals are closed under inclusion. We give multiple characterizations of Elias ideals and concrete criteria to identify them. We connect Elias ideals to other well-studied definitions: Ulrich, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathfrak {m}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">m</mi> </math></EquationSource> </InlineEquation>-full, integrally closed, trace ideals, etc. Applications are given regarding canonical ideals, conductors and the Auslander index.</p>

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Elias Ideals

  • 期刊论文

  • 发表日期：

    2025年11月3日

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  • Hailong Dao

摘要

Let \((R, \mathfrak {m})\) ( R , m ) be an one-dimensional local Cohen–Macaulay ring. An \(\mathfrak m\) m -primary ideal I of R is Elias if the types of I and of R/I are equal. Canonical and principal ideals are Elias, and Elias ideals are closed under inclusion. We give multiple characterizations of Elias ideals and concrete criteria to identify them. We connect Elias ideals to other well-studied definitions: Ulrich, \(\mathfrak {m}\) m -full, integrally closed, trace ideals, etc. Applications are given regarding canonical ideals, conductors and the Auslander index.