关于$\aleph_{0}$ -内射环的注记
A Note on $\aleph_{0}$-injective Rings
摘要：
环$R$称为右$\aleph_{0}$ -内射，如果从$R$的一个可数生成右理想到$R_{R}$的每个同态都可以扩张为从$R_{R}$到$R_{R}$的同态.本文给出了$\aleph_{0}$ -内射环的一些刻划。证明了若$R$是半局部的，则$R$是右$\aleph_{0}$ -内射的充要条件是$R$的可数生成小右理想到$R_{R}$的每个同态都可以扩张为从$R_{R}$到$R_{R}$的同态.还证明了若$R$是右诺特左$\aleph_{0}$ -内射的，则$R$是\emph{QF} .这一结果可以被认为是对Faith-Menal猜想的一种探讨。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
