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Keywords:
character modules; flat modules; cotorsion pairs
character modules; flat modules; cotorsion pairs
Summary:
Let $R$ be a ring. A left $R$-module $M$ is called an FC-module if $M^{+}= \operatorname{Hom}_{\mathbb{Z}}(M, \mathbb{Q}/\mathbb{Z})$ is a flat right $R$-module. In this paper, some homological properties of FC-modules are given. Let $n$ be a nonnegative integer and $\mathcal{FC}_{n}$ the class of all left $R$-modules $M$ such that the flat dimension of $M^{+}$ is less than or equal to $n$. It is shown that $({^{\bot}(\mathcal{FC}_{n}^{\bot})}, \mathcal{FC}_{n}^{\bot})$ is a complete cotorsion pair and if $R$ is a ring such that $\operatorname{fd}(({_RR})^{+})\leq n$ and $\mathcal{FC}_{n}$ is closed under direct sums, then $(\mathcal{FC}_{n}, \mathcal{FC}_{n}^{\bot})$ is a perfect cotorsion pair. In particular, some known results are obtained as corollaries.
References:
[1] Anderson F.W., Fuller K.R.: Rings and Categories of Modules. 2nd ed., Graduate Texts in Mathematics, 13, Springer, New York, 1992. MR 1245487 | Zbl 0765.16001
[2] Asensio Mayor J., Martinez Hernandez J.: Flat envelopes in commutative rings. Israel J. Math. 62 (1988), no. 1, 123--128. DOI 10.1007/BF02767358 | MR 0947834 | Zbl 0658.13010
[3] Asensio Mayor J., Martinez Hernandez J.: Monomorphic flat envelopes in commutative rings. Arch. Math. (Basel) 54 (1990), no. 5, 430--435. DOI 10.1007/BF01188669 | MR 1049197 | Zbl 0658.13010
[4] Cheatham T.J., Stone D.R.: Flat and projective character modules. Proc. Amer. Math. Soc. 81 (1981), no. 2, 175--177. DOI 10.1090/S0002-9939-1981-0593450-2 | MR 0593450 | Zbl 0458.16014
[5] Colby R.R.: Rings which have flat injective modules. J. Algebra 35 (1975), 239--252. DOI 10.1016/0021-8693(75)90049-6 | MR 0376763 | Zbl 0306.16015
[6] Couchot F.: Exemples d'anneaux auto-fp-injectifs. Comm. Algebra 10 (1982), no. 4, 339--360. DOI 10.1080/00927878208822720 | MR 0649340 | Zbl 0484.16010
[7] Ding N.Q., Chen J.L.: The flat dimensions of injective modules. Manuscripta Math. 78 (1993), 165--177. DOI 10.1007/BF02599307 | MR 1202159 | Zbl 0804.16005
[8] Eklof P.C., Trlifaj J.: How to make Ext vanish. Bull. London Math. Soc. 33 (2001), no. 1, 41--51. DOI 10.1112/blms/33.1.41 | MR 1798574 | Zbl 1030.16004
[9] Enochs E.E., Jenda O.M.G.: Relative Homological Algebra. de Gruyter Expositions in Mathematics, 30, de Gruyter, Berlin, 2000. MR 1753146 | Zbl 0952.13001
[10] Enochs E.E., Jenda O.M.G., Xu J.: The existence of envelopes. Rend. Sem. Mat. Univ. Padova 90 (1990), 45--51. MR 1257131 | Zbl 0803.16001
[11] Fieldhouse D.J.: Character modules. Comment. Math. Helv. 46 (1971), 274-276. DOI 10.1007/BF02566844 | MR 0294408 | Zbl 0258.16028
[12] Göbel R., Trlifaj J.: Approximations and Endomorphism Algebras of Modules. de Gruyter Expositions in Mathematics, 41, de Gruyter, Berlin, 2006. MR 2251271
[13] Holm H., Jøgensen P.: Covers, preenvelopes, and purity. Illinois J. Math. 52 (2008), 691--703. MR 2524661
[14] Jain S.: Flat and FP-injectivity. Proc. Amer. Math. Soc. 41 (1973), no. 2, 437--442. DOI 10.1090/S0002-9939-1973-0323828-9 | MR 0323828 | Zbl 0246.16013
[15] Mao L.X., Ding N.Q.: Envelopes and covers by modules of finite FP-injective and flat dimensions. Comm. Algebra 35 (2007), 835--849. MR 2305235 | Zbl 1122.16007
[16] Ramamurthi V.S.: On modules with projective character modules. Math. Japon. 23 (1978), 181--184. MR 0517797 | Zbl 0412.16015
[17] Rada J., Saorín M.: Rings characterized by (pre)envelopes and (pre)covers of their modules. Comm. Algebra 26 (1998), 899--912. DOI 10.1080/00927879808826172 | MR 1606190 | Zbl 0908.16003
[18] Stenström B.: Coherent rings and FP-injective modules. J. London Math. Soc. 2 (1970), 323--329. DOI 10.1112/jlms/s2-2.2.323 | MR 0258888
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