Rings whose nonsingular right modules are R -projective

Yusuf Alagöz; Sinem Benli; Engin Büyükaşık

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Volume: 62, Issue: 4, page 393-407
  • ISSN: 0010-2628

Abstract

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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 Σ - C S 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 ⊕ 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 𝒩 ⊥ of the class 𝒩 of nonsingular right modules.

How to cite

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Alagöz, Yusuf, Benli, Sinem, and Büyükaşık, Engin. "Rings whose nonsingular right modules are $R$-projective." Commentationes Mathematicae Universitatis Carolinae 62.4 (2021): 393-407. <http://eudml.org/doc/297492>.

@article{Alagöz2021,
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.},
author = {Alagöz, Yusuf, Benli, Sinem, Büyükaşık, Engin},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonsingular module; $R$-projective module; flat module; perfect ring},
language = {eng},
number = {4},
pages = {393-407},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Rings whose nonsingular right modules are $R$-projective},
url = {http://eudml.org/doc/297492},
volume = {62},
year = {2021},
}

TY - JOUR
AU - Alagöz, Yusuf
AU - Benli, Sinem
AU - Büyükaşık, Engin
TI - Rings whose nonsingular right modules are $R$-projective
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62
IS - 4
SP - 393
EP - 407
AB - 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.
LA - eng
KW - nonsingular module; $R$-projective module; flat module; perfect ring
UR - http://eudml.org/doc/297492
ER -

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