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<p>We study the freeness of the group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{Inv}(D)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Inv</mtext> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of invertible ideals of an integral domain <i>D</i>, and the freeness of some related groups of (fractional) ideals. We study the relation between <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textrm{Inv}(D)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Inv</mtext> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textrm{Inv}(D_P)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Inv</mtext> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mi>P</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, in particular in the locally finite case, and we analyze in more detail the case where <i>D</i> is Noetherian (obtaining a characterization of when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textrm{Inv}(D)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Inv</mtext> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is free for one-dimensional analytically unramified Noetherian domains) and where <i>D</i> is Prüfer.</p>

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Free groups of ideals

  • 期刊论文

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  • 发表日期：

    2025年11月8日

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  • Dario Spirito

摘要

We study the freeness of the group \(\textrm{Inv}(D)\) Inv ( D ) of invertible ideals of an integral domain D, and the freeness of some related groups of (fractional) ideals. We study the relation between \(\textrm{Inv}(D)\) Inv ( D ) and \(\textrm{Inv}(D_P)\) Inv ( D P ) , in particular in the locally finite case, and we analyze in more detail the case where D is Noetherian (obtaining a characterization of when \(\textrm{Inv}(D)\) Inv ( D ) is free for one-dimensional analytically unramified Noetherian domains) and where D is Prüfer.