| Title:
|
Rings whose nonsingular right modules are $R$-projective (English) |
| Author:
|
Alagöz, Yusuf |
| Author:
|
Benli, Sinem |
| Author:
|
Büyükaşık, Engin |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
62 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
393-407 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
nonsingular module |
| Keyword:
|
$R$-projective module |
| Keyword:
|
flat module |
| Keyword:
|
perfect ring |
| MSC:
|
16D10 |
| MSC:
|
16D40 |
| MSC:
|
16D80 |
| MSC:
|
16E30 |
| idZBL:
|
Zbl 07511568 |
| idMR:
|
MR4405811 |
| DOI:
|
10.14712/1213-7243.2021.036 |
| . |
| Date available:
|
2022-02-21T13:19:40Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149363 |
| . |
| Reference:
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