| Title:
|
Coherence relative to a weak torsion class (English) |
| Author:
|
Zhu, Zhanmin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
455-474 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a ring. A subclass $\mathcal {T}$ of left $R$-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let $\mathcal {T}$ be a weak torsion class of left $R$-modules and $n$ a positive integer. Then a left $R$-module $M$ is called $\mathcal {T}$-finitely generated if there exists a finitely generated submodule $N$ such that $M/N\in \mathcal {T}$; a left $R$-module $A$ is called $(\mathcal {T},n)$-presented if there exists an exact sequence of left $R$-modules $$ 0\longrightarrow K_{n-1}\longrightarrow F_{n-1}\longrightarrow \cdots \longrightarrow F_1\longrightarrow F_0\longrightarrow M\longrightarrow 0 $$ such that $F_0,\cdots ,F_{n-1}$ are finitely generated free and $K_{n-1}$ is $\mathcal {T}$-finitely generated; a left $R$-module $M$ is called $(\mathcal {T},n)$-injective, if ${\rm Ext}^n_R(A, M)=0$ for each $(\mathcal {T},n+1)$-presented left $R$-module $A$; a right $R$-module $M$ is called $(\mathcal {T},n)$-flat, if ${\rm Tor}^R_n(M, A)=0$ for each $(\mathcal {T},n+1)$-presented left $R$-module $A$. A ring $R$ is called $(\mathcal {T},n)$-coherent, if every $(\mathcal {T},n+1)$-presented module is $(n+1)$-presented. Some characterizations and properties of these modules and rings are given. (English) |
| Keyword:
|
$(\mathcal {T},n)$-presented module |
| Keyword:
|
$(\mathcal {T},n)$-injective module |
| Keyword:
|
$(\mathcal {T},n)$-flat module |
| Keyword:
|
$(\mathcal {T},n)$-coherent ring |
| MSC:
|
16D40 |
| MSC:
|
16D50 |
| MSC:
|
16P70 |
| idZBL:
|
Zbl 06890383 |
| idMR:
|
MR3819184 |
| DOI:
|
10.21136/CMJ.2018.0494-16 |
| . |
| Date available:
|
2018-06-11T10:55:10Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147229 |
| . |
| Reference:
|
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