Let R and S be rings and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1837_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$_R\omega _S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>R</mi> <mrow /> </mmultiscripts> <msub> <mi>ω</mi> <mi>S</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> a semidualizing bimodule, and let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1837_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\geqslant 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We characterize the extension closure of the category of adjoint k-cotorsionfree modules with respect to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1837_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1837_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$1\leqslant k\leqslant n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>k</mi> <mo>⩽</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> in terms of the (strong) cograde conditions of certain modules.

Extension Closure of Adjoint Cotorsionfree Modules

期刊论文
发表日期：

2025年2月25日

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Xi Tang,
Zhaoyong Huang

摘要

Let R and S be rings and $_R\omega _S$ $_{R} ω_{S}$ a semidualizing bimodule, and let $n\geqslant 1$ $n ⩾ 1$ . We characterize the extension closure of the category of adjoint k-cotorsionfree modules with respect to $\omega $ $ω$ for any $1\leqslant k\leqslant n$ $1 ⩽ k ⩽ n$ in terms of the (strong) cograde conditions of certain modules.