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<p>Let <i>R</i> ⋉ <i>M</i> be a trivial extension of a ring <i>R</i> by an <i>R-R</i>-bimodule <i>M</i>. Sufficient and necessary conditions are established for projectively coresolved Gorenstein flat (PGF, for short) modules over <i>R</i> ⋉ <i>M</i>. More pecisely, it is proved that (<i>X, α</i>) is a PGF left <i>R</i> ⋉ <i>M</i>-module if and only if Coker(<i>α</i>) is a PGF left <i>R</i>-module and the sequence <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10587_2025_523_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="272" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\otimes_{R}M\otimes_{R}X \overset{M\otimes\alpha} \longrightarrow M\otimes_{R}X \overset{\alpha} \longrightarrow X\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>M</mi> <msub> <mo>⊗</mo> <mrow> <mi>R</mi> </mrow> </msub> <mi>M</mi> <msub> <mo>⊗</mo> <mrow> <mi>R</mi> </mrow> </msub> <mi>X</mi> <mover> <mo stretchy="false">⟶</mo> <mrow> <mi>M</mi> <mo>⊗</mo> <mi>α</mi> </mrow> </mover> <mi>M</mi> <msub> <mo>⊗</mo> <mrow> <mi>R</mi> </mrow> </msub> <mi>X</mi> <mover> <mo stretchy="false">⟶</mo> <mi>α</mi> </mover> <mi>X</mi> </math></EquationSource> </InlineEquation> is exact under some assumptions on <i>M</i>. As applications, it is characterized PGF modules over Morita rings with zero bimodule homomorphisms.</p>

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Projectively coresolved Gorenstein flat modules over trivial ring extensions

  • 期刊论文

  • 发表日期：

    2025年2月18日

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  • Zhanping Wang,
  • Yuanhui Jin,
  • Jianyuan He

摘要

Let R ⋉ M be a trivial extension of a ring R by an R-R-bimodule M. Sufficient and necessary conditions are established for projectively coresolved Gorenstein flat (PGF, for short) modules over R ⋉ M. More pecisely, it is proved that (X, α) is a PGF left R ⋉ M-module if and only if Coker(α) is a PGF left R-module and the sequence \(M\otimes_{R}M\otimes_{R}X \overset{M\otimes\alpha} \longrightarrow M\otimes_{R}X \overset{\alpha} \longrightarrow X\) M ⊗ R M ⊗ R X ⟶ M ⊗ α M ⊗ R X ⟶ α X is exact under some assumptions on M. As applications, it is characterized PGF modules over Morita rings with zero bimodule homomorphisms.