# Institute of Mathematics

## 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:  Institute of Mathematics CAS

References:
[1] S. U. Chase: Direct products of modules. Trans. Am. Math. Soc. 97 (1960), 457-473. DOI 10.1090/S0002-9947-1960-0120260-3 | MR 0120260 | Zbl 0100.26602
[2] T. J. Cheatham, D. R. Stone: Flat and projective character modules. Proc. Am. Math. Soc. 81 (1981), 175-177. DOI 10.1090/S0002-9939-1981-0593450-2 | MR 0593450 | Zbl 0458.16014
[3] J. Chen, N. Ding: On $n$-coherent rings. Commun. Algebra 24 (1996), 3211-3216. DOI 10.1080/00927879608825742 | MR 1402554 | Zbl 0877.16010
[4] D. L. Costa: Parameterizing families of non-Noetherian rings. Commun. Algebra 22 (1994), 3997-4011. DOI 10.1080/00927879408825061 | MR 1280104 | Zbl 0814.13010
[5] E. Enochs: A note on absolutely pure modules. Canad. Math. Bull. 19 (1976), 361-362. DOI 10.4153/CMB-1976-054-5 | MR 0429988 | Zbl 0346.16020
[6] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
[7] E. E. Enochs, O. M. G. Jenda, J. A. Lopez-Ramos: The existence of Gorenstein flat covers. Math. Scand. 94 (2004), 46-62. DOI 10.7146/math.scand.a-14429 | MR 2032335 | Zbl 1061.16003
[8] M. Finkel Jones: Coherence relative to an hereditary torsion theory. Commun. Algebra 10 (1982), 719-739. DOI 10.1080/00927878208822745 | MR 0650869 | Zbl 0483.16027
[9] H. Holm, P. Jørgensen: Covers, precovers, and purity. Illinois J. Math. 52 (2008), 691-703. MR 2524661 | Zbl 1189.16007
[10] L. Mao, N. Ding: Relative coherence of rings. J. Algebra Appl. 11 (2012), 1250047, 16 pages. DOI 10.1142/S0219498811005749 | MR 2928115 | Zbl 1252.16018
[11] C. Megibben: Absolutely pure modules. Proc. Am. Math. Soc. 26 (1970), 561-566. DOI 10.1090/S0002-9939-1970-0294409-8 | MR 0294409 | Zbl 0216.33803
[12] J. J. Rotman: An Introduction to Homological Algebra. Pure and Applied Mathematics 85, Academic Press, Harcourt Brace Jovanovich Publishers, New York-London (1979). MR 0538169 | Zbl 0441.18018
[13] B. Stenström: Coherent rings and FP-injective modules. J. Lond. Math. Soc., II. Ser. 2 (1970), 323-329. DOI 10.1112/jlms/s2-2.2.323 | MR 0271145 | Zbl 0194.06602
[14] B. Stenström: Rings of Quotients. An Introduction to Methods of Ring Theory. Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer, New York (1975). DOI 10.1007/978-3-642-66066-5 | MR 0389953 | Zbl 0296.16001
[15] J. Trlifaj: Cover, Envelopes, and Cotorsion Theories. Lecture notes for the workshop. Homological Methods in Module Theory, Cortona (2000).
[16] R. Wisbauer: Foundations of Module and Ring Theory. A Handbook for Study and Research. Algebra, Logic and Applications 3, Gordon and Breach Science Publishers, Philadelphia (1991). MR 1144522 | Zbl 0746.16001
[17] X. Yang, Z. Liu: $n$-flat and $n$-FP-injective modules. Czech. Math. J. 61 (2011), 359-369. DOI 10.1007/s10587-011-0080-4 | MR 2905409 | Zbl 1249.13011
[18] D. Zhou: On $n$-coherent rings and $(n,d)$-rings. Commun. Algebra 32 (2004), 2425-2441. DOI 10.1081/AGB-120037230 | MR 2100480 | Zbl 1089.16001

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

PDF available at: