<p>In this paper, we introduce the notion of generalized <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {W}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">W</mi> </math></EquationSource> </InlineEquation>-Gorenstein modules respect to some subclass <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {W}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">W</mi> </math></EquationSource> </InlineEquation>, extending the classical notion of Gorenstein projective modules. By exploiting the correspondence between projective modules over the endomorphism ring <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{End}_R(C)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>End</mtext> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a module <i>C</i> and elements of its additive closure <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {W}=\textrm{Add}_R(C)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">W</mi> <mo>=</mo> <msub> <mtext>Add</mtext> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we establish a fundamental correspondence between Gorenstein projective <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{End}_R(C)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>End</mtext> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-modules and generalized <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textrm{Add}_R(C)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Add</mtext> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-Gorenstein modules. This result refines existing relative homological settings and provides a natural extension of well-known results in Gorenstein homological algebra. We explore key properties, such as closure under direct summands and sums, and identify conditions under which the class of generalized <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {W}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">W</mi> </math></EquationSource> </InlineEquation>-Gorenstein modules coincides with other classes of modules, like Gorenstein projective modules.</p>

Generalized $\mathcal {W}$-Gorenstein Modules

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2026年7月20日

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Driss Bennis,
Christian Lomp,
Abderrazak Nassir

摘要

In this paper, we introduce the notion of generalized $\mathcal {W}$ $W$ -Gorenstein modules respect to some subclass $\mathcal {W}$ $W$ , extending the classical notion of Gorenstein projective modules. By exploiting the correspondence between projective modules over the endomorphism ring $\textrm{End}_R(C)$ ${End}_{R} (C)$ of a module C and elements of its additive closure $\mathcal {W}=\textrm{Add}_R(C)$ $W = {Add}_{R} (C)$ , we establish a fundamental correspondence between Gorenstein projective $\textrm{End}_R(C)$ ${End}_{R} (C)$ -modules and generalized $\textrm{Add}_R(C)$ ${Add}_{R} (C)$ -Gorenstein modules. This result refines existing relative homological settings and provides a natural extension of well-known results in Gorenstein homological algebra. We explore key properties, such as closure under direct summands and sums, and identify conditions under which the class of generalized $\mathcal {W}$ $W$ -Gorenstein modules coincides with other classes of modules, like Gorenstein projective modules.

Generalized \(\mathcal {W}\)-Gorenstein Modules

摘要