A Note on ${\mathrm{\aleph }}_0$-injective Rings

A ring $R$ is called right ${\mathrm{\aleph }}_0$-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 ${\mathrm{\aleph }}_0$-injective rings are given. It is proved that if $R$ is semilocal, then $R$ is right ${\mathrm{\aleph }}_0$-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 ${\mathrm{\aleph }}_0$-injective, then $R$ is \emph{QF}. This result can be considered as an approach to the Faith-Menal conjecture.