# Institute of Mathematics

## 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.
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
Classification MSC:  16D40, 16D50, 16E60, 16P70
DOI:  10.21136/CMJ.2020.0377-18

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Affiliations:   Zhanmin Zhu, Jiaxing University, 56 South Yuexiu Road, Nanhu, Jiaxing, 314001 Zhejiang, P. R. China, e-mail: zhuzhanminzjxu@hotmail.com

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