Duality of Preenvelopes and Pure Injective Modules
Duality of Preenvelopes and Pure Injective Modules
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DOI：
10.4153/cmb-2013-023-x
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发表时间：
2013-06
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影响因子：
--
通讯作者：
Zhaoyong Huang
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文献类型：
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作者：
Zhaoyong Huang
Abstract 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.
Abstract 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.