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Duality of Preenvelopes and Pure Injective Modules
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 2 / 14 June 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 318-325
- Print publication:
- 14 June 2014
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Let
$R$ be an arbitrary ring and let 
${{\left( - \right)}^{+}}\,=\,\text{Ho}{{\text{m}}_{\mathbb{Z}}}\left( -,\,{\mathbb{Q}}/{\mathbb{Z}}\; \right)$, where 
$\mathbb{Z}$ is the ring of integers and 
$\mathbb{Q}$ is the ring of rational numbers. Let 
$\mathcal{C}$ be a subcategory of left $R$-modules and 
$\mathcal{D}$ a subcategory of right $R$-modules such that 
${{X}^{+}}\,\in \,\mathcal{D}$ for any 
$X\,\in \,\mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism 
$f:\,A\to \,C$ of left $R$-modules with 
$C\,\in \,\mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of 
$A$ provided 
${{f}^{+}}:\,{{C}^{+}}\,\to \,{{A}^{+}}$ is a $\mathcal{D}$-(pre)cover of 
${{A}^{+}}$. Some applications of this result are given.
Cotorsion pairs and model structures on Ch(R)
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 54 / Issue 3 / October 2011
- Published online by Cambridge University Press:
- 17 August 2011, pp. 783-797
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We show that if the given cotorsion pair
in the category of modules is complete and hereditary, then both of the induced cotorsion pairs in the category of complexes are complete. We also give a cofibrantly generated model structure that can be regarded as a generalization of the projective model structure.
in the category of modules is complete and hereditary, then both of the induced cotorsion pairs in the category of complexes are complete. We also give a cofibrantly generated model structure that can be regarded as a generalization of the projective model structure.