Rings whose nonsingular right modules are $R$-projective

Abstract

A right $R$-module $M$ is called $R$-projective provided that it is projective relative to the right $R$-module $R_{R}$. This paper deals with the rings whose all nonsingular right modules are $R$-projective. For a right nonsingular ring $R$, we prove that $R_{R}$ is of finite Goldie rank and all nonsingular right $R$-modules are $R$-projective if and only if $R$ is right finitely $\Sigma$-$CS$ and flat right $R$-modules are $R$-projective. Then, $R$-projectivity of the class of nonsingular injective right modules is also considered. Over right nonsingular rings of finite right Goldie rank, it is shown that $R$-projectivity of nonsingular injective right modules is equivalent to $R$-projectivity of the injective hull $E(R_{R})$. In this case, the injective hull $E(R_{R})$ has the decomposition $E(R_{R})=U_{R} \oplus V_{R}$, where $U$ is projective and $\operatorname{Hom}(V,R/I)=0$ for each right ideal $I$ of $R$. Finally, we focus on the right orthogonal class $\mathcal{N}^{\perp}$ of the class $\mathcal{N}$ of nonsingular right modules.