Czechoslovak Mathematical Journal, Vol. 70, No. 3, pp. 657-674, 2020
Strongly $(\mathcal{T},n)$-coherent rings, $(\mathcal{T},n)$-semihereditary rings and $(\mathcal{T},n)$-regular rings
Zhanmin Zhu
Received August 13, 2018. Published online July 7, 2020.
Abstract: 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 $0\rightarrow K\rightarrow P\rightarrow C\rightarrow0$ 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 $0\rightarrow K\rightarrow P\rightarrow C\rightarrow0$ 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.