J-regular rings with injectivities

TL;DR

This paper studies J-regular rings, showing their properties related to right injectivity and extending known results about their structure and injective modules.

Contribution

It establishes new characterizations of right n-injectivity and FP-injectivity in J-regular rings, improving existing results.

Findings

R is right n-injective iff certain homomorphisms extend

R is right FP-injective iff it is right (J, R)-FP-injective

Improves known results on J-regular rings

Abstract

A ring $R$ is called a J-regular ring if R/J(R) is von Neumann regular, where J(R) is the Jacobson radical of R. It is proved that if R is J-regular, then (i) R is right n-injective if and only if every homomorphism from an $n$-generated small right ideal of $R$ to $R_{R}$ can be extended to one from $R_{R}$ to $R_{R}$; (ii) R is right FP-injective if and only if R is right (J, R)-FP-injective. Some known results are improved.