FC-modules with an application to cotorsion pairs

Yonghua Guo

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 4, page 513-519
  • ISSN: 0010-2628

Abstract

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Let R be a ring. A left R -module M is called an FC-module if M + = Hom ℤ ( M , ℚ / 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 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 ( ⊥ ( ℱ𝒞 n ⊥ ) , ℱ𝒞 n ⊥ ) is a complete cotorsion pair and if R is a ring such that fd ( ( R R ) + ) ≤ n and ℱ𝒞 n is closed under direct sums, then ( ℱ𝒞 n , ℱ𝒞 n ⊥ ) is a perfect cotorsion pair. In particular, some known results are obtained as corollaries.

How to cite

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Guo, Yonghua. "FC-modules with an application to cotorsion pairs." Commentationes Mathematicae Universitatis Carolinae 50.4 (2009): 513-519. <http://eudml.org/doc/35126>.

@article{Guo2009,
abstract = {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 $\operatorname\{fd\}((\{_RR\})^\{+\})\le n$ 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.},
author = {Guo, Yonghua},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {character modules; flat modules; cotorsion pairs; FC-modules; character modules; flat modules; cotorsion pairs; flat dimension},
language = {eng},
number = {4},
pages = {513-519},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {FC-modules with an application to cotorsion pairs},
url = {http://eudml.org/doc/35126},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Guo, Yonghua
TI - FC-modules with an application to cotorsion pairs
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 4
SP - 513
EP - 519
AB - 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 $\operatorname{fd}(({_RR})^{+})\le n$ 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.
LA - eng
KW - character modules; flat modules; cotorsion pairs; FC-modules; character modules; flat modules; cotorsion pairs; flat dimension
UR - http://eudml.org/doc/35126
ER -

References

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  1. Anderson F.W., Fuller K.R., Rings and Categories of Modules, 2nd ed., Graduate Texts in Mathematics, 13, Springer, New York, 1992. Zbl0765.16001MR1245487
  2. Asensio Mayor J., Martinez Hernandez J., 10.1007/BF02767358, Israel J. Math. 62 (1988), no. 1, 123--128. Zbl0658.13010MR0947834DOI10.1007/BF02767358
  3. Asensio Mayor J., Martinez Hernandez J., 10.1007/BF01188669, Arch. Math. (Basel) 54 (1990), no. 5, 430--435. Zbl0658.13010MR1049197DOI10.1007/BF01188669
  4. Cheatham T.J., Stone D.R., 10.1090/S0002-9939-1981-0593450-2, Proc. Amer. Math. Soc. 81 (1981), no. 2, 175--177. Zbl0458.16014MR0593450DOI10.1090/S0002-9939-1981-0593450-2
  5. Colby R.R., 10.1016/0021-8693(75)90049-6, J. Algebra 35 (1975), 239--252. Zbl0306.16015MR0376763DOI10.1016/0021-8693(75)90049-6
  6. Couchot F., 10.1080/00927878208822720, Comm. Algebra 10 (1982), no. 4, 339--360. Zbl0484.16010MR0649340DOI10.1080/00927878208822720
  7. Ding N.Q., Chen J.L., 10.1007/BF02599307, Manuscripta Math. 78 (1993), 165--177. Zbl0804.16005MR1202159DOI10.1007/BF02599307
  8. Eklof P.C., Trlifaj J., 10.1112/blms/33.1.41, Bull. London Math. Soc. 33 (2001), no. 1, 41--51. Zbl1030.16004MR1798574DOI10.1112/blms/33.1.41
  9. Enochs E.E., Jenda O.M.G., Relative Homological Algebra, de Gruyter Expositions in Mathematics, 30, de Gruyter, Berlin, 2000. Zbl0952.13001MR1753146
  10. Enochs E.E., Jenda O.M.G., Xu J., The existence of envelopes, Rend. Sem. Mat. Univ. Padova 90 (1990), 45--51. Zbl0803.16001MR1257131
  11. Fieldhouse D.J., 10.1007/BF02566844, Comment. Math. Helv. 46 (1971), 274-276. Zbl0258.16028MR0294408DOI10.1007/BF02566844
  12. Göbel R., Trlifaj J., Approximations and Endomorphism Algebras of Modules, de Gruyter Expositions in Mathematics, 41, de Gruyter, Berlin, 2006. MR2251271
  13. Holm H., Jøgensen P., Covers, preenvelopes, and purity, Illinois J. Math. 52 (2008), 691--703. MR2524661
  14. Jain S., 10.1090/S0002-9939-1973-0323828-9, Proc. Amer. Math. Soc. 41 (1973), no. 2, 437--442. Zbl0246.16013MR0323828DOI10.1090/S0002-9939-1973-0323828-9
  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. Zbl1122.16007MR2305235
  16. Ramamurthi V.S., On modules with projective character modules, Math. Japon. 23 (1978), 181--184. Zbl0412.16015MR0517797
  17. Rada J., Saorín M., 10.1080/00927879808826172, Comm. Algebra 26 (1998), 899--912. Zbl0908.16003MR1606190DOI10.1080/00927879808826172
  18. Stenström B., 10.1112/jlms/s2-2.2.323, J. London Math. Soc. 2 (1970), 323--329. MR0258888DOI10.1112/jlms/s2-2.2.323

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