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Keywords:
$(\mathcal {T},n)$-presented module; $(\mathcal {T},n)$-injective module; $(\mathcal {T},n)$-flat module; $(\mathcal {T},n)$-coherent ring
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.
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