Coherence relative to a weak torsion class

Zhu, Zhanmin. "Coherence relative to a weak torsion class." Czechoslovak Mathematical Journal 68.2 (2018): 455-474. <http://eudml.org/doc/294171>.

@article{Zhu2018,
abstract = {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.},
author = {Zhu, Zhanmin},
journal = {Czechoslovak Mathematical Journal},
keywords = {$(\mathcal \{T\},n)$-presented module; $(\mathcal \{T\},n)$-injective module; $(\mathcal \{T\},n)$-flat module; $(\mathcal \{T\},n)$-coherent ring},
language = {eng},
number = {2},
pages = {455-474},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Coherence relative to a weak torsion class},
url = {http://eudml.org/doc/294171},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Zhu, Zhanmin
TI - Coherence relative to a weak torsion class
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 2
SP - 455
EP - 474
AB - 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.
LA - eng
KW - $(\mathcal {T},n)$-presented module; $(\mathcal {T},n)$-injective module; $(\mathcal {T},n)$-flat module; $(\mathcal {T},n)$-coherent ring
UR - http://eudml.org/doc/294171
ER -