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<p>Let <i>R</i> be a commutative Noetherian ring (with identity) and <i>M</i>, <i>A</i> be two non-zero <i>R</i>-modules. In this paper, it is shown that every proper submodule of <i>M</i> has a primary decomposition if and only if <i>M</i> possesses a finitely generated submodule <i>N</i> such that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\dim R/(N:_RM)=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>dim</mo> <mi>R</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>N</mi> <msub> <mo>:</mo> <mi>R</mi> </msub> <mi>M</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Also, it is shown that every submodule of <i>A</i> has a secondary representation if and only if <i>A</i> has a submodule <i>B</i> such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\dim R/(0:_RB)=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>dim</mo> <mi>R</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mn>0</mn> <msub> <mo>:</mo> <mi>R</mi> </msub> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and the <i>R</i>-module <i>A</i>/<i>B</i> is Artinian.</p>

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On the Primary Decomposition and Secondary Representation of Modules

  • 期刊论文

  • 发表日期：

    2025年7月28日

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  • Kamal Bahmanpour

摘要

Let R be a commutative Noetherian ring (with identity) and M, A be two non-zero R-modules. In this paper, it is shown that every proper submodule of M has a primary decomposition if and only if M possesses a finitely generated submodule N such that \(\dim R/(N:_RM)=0\) dim R / ( N : R M ) = 0 . Also, it is shown that every submodule of A has a secondary representation if and only if A has a submodule B such that \(\dim R/(0:_RB)=0\) dim R / ( 0 : R B ) = 0 and the R-module A/B is Artinian.