Czechoslovak Mathematical Journal, first online, pp. 1-18


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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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] J. Chen, N. Ding: A note on existence of envelopes and covers. Bull. Aust. Math. Soc. 54 (1996), 383-390. DOI 10.1017/S0004972700021791 | MR 1419601 | Zbl 0882.16002
[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. 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
[6] E. E. Enochs, O. M. G. Jenda, J. A. López-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
[7] S. Jain: Flat and FP-injectivity. Proc. Am. Math. Soc. 41 (1973), 437-442. DOI 10.1090/S0002-9939-1973-0323828-9 | MR 0323828 | Zbl 0246.16013
[8] S.-E. Kabbaj, N. Mahdou: Trivial extensions defined by coherent-like conditions. Commun. Algebra 32 (2004), 3937-3953. DOI 10.1081/AGB-200027791 | MR 2097439 | Zbl 1068.13002
[9] L. Mao, N. Ding: FP-projective dimensions. Commun. Algebra 33 (2005), 1153-1170. DOI 10.1081/AGB-200053832 | MR 2136693 | Zbl 1097.16005
[10] 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
[11] 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 0258888 | Zbl 0194.06602
[12] J. Trlifaj: Cover, Envelopes, and Cotorsion Theories. Lecture Notes for the Workshop "Homological Methods in Module Theory" Cortona, September 10-16 (2000).
[13] 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
[14] Z. Zhu: On $n$-coherent rings, $n$-hereditary rings and $n$-regular rings. Bull. Iran. Math. Soc. 37 (2011), 251-267. MR 2915464 | Zbl 1277.16007
[15] Z. Zhu: Some results on $(n,d)$-injective modules, $(n,d)$-flat modules and $n$-coherent rings. Comment. Math. Univ. Carol. 56 (2015), 505-513. DOI 10.14712/1213-7243.2015.133 | MR 3434225 | Zbl 1363.16013
[16] Z. Zhu: Coherence relative to a weak torsion class. Czech. Math. J. 68 (2018), 455-474. DOI 10.21136/CMJ.2018.0494-16 | MR 3819184 | Zbl 06890383

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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