## Yusuf Alagöz, Sinem Benli, Engin Büyükaşik

*Rings whose nonsingular right modules are R-projective*

Comment.Math.Univ.Carolin. 62,4 (2021) 393-407.**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 {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.

**Keywords:** nonsingular module; R-projective module; flat module; perfect ring

**DOI:** DOI 10.14712/1213-7243.2021.036

**AMS Subject Classification:** 16D10 16D40 16D80 16E30

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