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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
Summary:
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 $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ 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 $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ 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.
References:
[1] Chase, S. U.: 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] Chen, J., Ding, N.: 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] Chen, J., Ding, N.: On $n$-coherent rings. Commun. Algebra 24 (1996), 3211-3216. DOI 10.1080/00927879608825742 | MR 1402554 | Zbl 0877.16010
[4] Costa, D. L.: Parameterizing families of non-Noetherian rings. Commun. Algebra 22 (1994), 3997-4011. DOI 10.1080/00927879408825061 | MR 1280104 | Zbl 0814.13010
[5] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra. de Gruyter Expositions in Mathematics 30. Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
[6] Enochs, E. E., Jenda, O. M. G., López-Ramos, J. A.: 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] Jain, S.: 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] Kabbaj, S.-E., Mahdou, N.: Trivial extensions defined by coherent-like conditions. Commun. Algebra 32 (2004), 3937-3953. DOI 10.1081/AGB-200027791 | MR 2097439 | Zbl 1068.13002
[9] Mao, L., Ding, N.: FP-projective dimensions. Commun. Algebra 33 (2005), 1153-1170. DOI 10.1081/AGB-200053832 | MR 2136693 | Zbl 1097.16005
[10] Megibben, C.: 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] Stenström, B.: 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] Trlifaj, J.: Cover, Envelopes, and Cotorsion Theories. Lecture Notes for the Workshop ``Homological Methods in Module Theory'' Cortona, September 10-16 (2000).
[13] Zhou, D.: 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] Zhu, Z.: 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] Zhu, Z.: 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] Zhu, Z.: 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
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