Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 68, No. 2, pp. 455-474, 2018

Coherence relative to a weak torsion class

Zhanmin Zhu

Received September 22, 2016. First published February 10, 2018.

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\rarrow K_{n-1}\rarrow F_{n-1}\rarrow\cdots\rarrow F_1\rarrow F_0\rarrow M\rarrow0 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.

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

Classification MSC: 16D40, 16D50, 16P70

DOI: 10.21136/CMJ.2018.0494-16

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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Affiliations: Zhanmin Zhu, Department of Mathematics, Jiaxing University, 118 Jiahang Rd, Nanhu, 314001 Jiaxing, Zhejiang, P. R. China e-mail: zhuzhanminzjxu@hotmail.com

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