# Article

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