| Title:
|
Strongly $(\mathcal {T},n)$-coherent rings, $(\mathcal {T},n)$-semihereditary rings and $(\mathcal {T},n)$-regular rings (English) |
| Author:
|
Zhu, Zhanmin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
657-674 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal {T}$ be a weak torsion class of left $R$-modules and $n$ a positive integer. A left $R$-module $M$ is called $(\mathcal {T},n)$-injective if ${\rm Ext}^n_R(C, M)=0$ for each $(\mathcal {T},n+1)$-presented left $R$-module $C$; a right $R$-module $M$ is called $(\mathcal {T},n)$-flat if ${\rm Tor}^R_n(M, C)=0$ for each $(\mathcal {T},n+1)$-presented left $R$-module $C$; a left $R$-module $M$ is called $(\mathcal {T},n)$-projective if ${\rm Ext}^n_R(M, N)=0$ for each $(\mathcal {T},n)$-injective left $R$-module $N$; the ring $R$ is called strongly $(\mathcal {T},n)$-coherent if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C$ is $(\mathcal {T},n+1)$-presented and $P$ is finitely generated projective, then $K$ is $(\mathcal {T},n)$-projective; the ring $R$ is called $(\mathcal {T},n)$-semihereditary if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C$ is $(\mathcal {T},n+1)$-presented and $P$ is finitely generated projective, then ${\rm pd} (K)\leq n-1$. Using the concepts of $(\mathcal {T},n)$-injectivity and $(\mathcal {T},n)$-flatness of modules, we present some characterizations of strongly $(\mathcal {T},n)$-coherent rings, $(\mathcal {T},n)$-semihereditary rings and $(\mathcal {T},n)$-regular rings. (English) |
| Keyword:
|
$(\mathcal {T},n)$-injective module |
| Keyword:
|
$(\mathcal {T},n)$-flat module |
| Keyword:
|
strongly $(\mathcal {T},n)$-coherent ring |
| Keyword:
|
$(\mathcal {T},n)$-semihereditary ring |
| Keyword:
|
$(\mathcal {T},n)$-regular ring |
| MSC:
|
16D40 |
| MSC:
|
16D50 |
| MSC:
|
16E60 |
| MSC:
|
16P70 |
| idZBL:
|
07250681 |
| idMR:
|
MR4151697 |
| DOI:
|
10.21136/CMJ.2020.0377-18 |
| . |
| Date available:
|
2020-09-07T09:34:59Z |
| Last updated:
|
2022-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148320 |
| . |
| Reference:
|
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