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

TL;DR

This paper explores the properties and characterizations of right leph_{0}-injective rings, providing new criteria and linking these rings to well-known classes like QF rings, thus contributing to the understanding of their structure.

Contribution

It offers new characterizations of leph_{0}-injective rings, especially in semilocal and noetherian cases, and connects these properties to the Faith-Menal conjecture.

Findings

Semilocal leph_{0}-injective rings characterized by small right ideals

Right noetherian and left leph_{0}-injective rings are QF rings

Provides an approach to the Faith-Menal conjecture

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.