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arXiv · 0710.5565

A Note on $\aleph_{0}$-injective Rings

Liang Shen
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Abstract

A ring $R$ is called right $\aleph_{0}$-injective if every homomorphism from a countably generated right ideal of $R$ to $R_{R}$ can be extended to a homomorphism from $R_{R}$ to $R_{R}$. In this note, some characterizations of $\aleph_{0}$-injective rings are given. It is proved that if $R$ is semilocal, then $R$ is right $\aleph_{0}$-injective if and only if every homomorphism from a countably generated small right ideal of $R$ to $R_{R}$ can be extended to one from $R_{R}$ to $R_{R}$. It is also shown that if $R$ is right noetherian and left $\aleph_{0}$-injective, then $R$ is \emph{QF}. This result can be considered as an approach to the Faith-Menal conjecture.

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Liang Shen. 2010-05-23. A Note on $\aleph_{0}$-injective Rings. https://arxiv.org/abs/0710.5565

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