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Fuchs, László; Lee, Sang Bum

On modules over commutative rings. (English) Zbl 1407.13007

J. Aust. Math. Soc. 103, No. 3, 341-356 (2017).
The authors extend some results that have been proved for modules over integral domains to modules over arbitrary commutative rings with identity. In Sections 2 and 3, they discuss torsion-freeness and divisibility. In Sections 4 and 5, they provide application of Matlis-cotorsion modules in two category equivalences. Finally, the authors prove results on rings for which ring of quotients has projective dimension \(1\).
Reviewer: Avanish Kumar Chaturvedi (Allahabad)

Cited in 1 Review
Cited in 11 Documents

MSC:

13C13 Other special types of modules and ideals in commutative rings
13C11 Injective and flat modules and ideals in commutative rings

Keywords:

commutative rings; torsion-free; torsion; divisible and \(h\)-divisible modules; cotorsion modules; generalized Matlis category equivalence; tight system; Matlis rings
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Full Text: DOI

References:

[1] L.Angeleri Hügel, D.Herbera and J.Trlifaj, ‘Divisible modules and localization’, J. Algebra294 (2005), 519-551.10.1016/j.jalgebra.2005.03.019 · Zbl 1117.13014 · doi:10.1016/j.jalgebra.2005.03.019
[2] H.Bass, ‘Finitistic dimension and a homological characterization of semiprimary rings’, Trans. Amer. Math. Soc.95 (1960), 466-488.10.1090/S0002-9947-1960-0157984-8 · Zbl 0094.02201 · doi:10.1090/S0002-9947-1960-0157984-8
[3] S.Bazzoni, P. C.Eklof and J.Trlifaj, ‘Tilting cotorsion pairs’, Bull. Lond. Math. Soc.37 (2005), 683-696.10.1112/S0024609305004728 · Zbl 1098.16006 · doi:10.1112/S0024609305004728
[4] S.Bazzoni and D.Herbera, ‘Cotorsion pairs generated by modules of bounded projective dimension’, Israel J. Math.174 (2009), 119-160.10.1007/s11856-009-0106-x · Zbl 1232.16009 · doi:10.1007/s11856-009-0106-x
[5] L.Bican, R.El Bashir and E.Enochs, ‘All modules have flat covers’, Bull. Lond. Math. Soc.33 (2001), 385-390.10.1017/S0024609301008104 · Zbl 1029.16002 · doi:10.1017/S0024609301008104
[6] H.Cartan and S.Eilenberg, Homological Algebra (Princeton University Press, Princeton, NJ, 1956). · Zbl 0075.24305
[7] J.Dauns and L.Fuchs, ‘Torsion-freeness for rings with zero-divisors’, J. Algebra Appl.3 (2004), 221-237.10.1142/S0219498804000885 · Zbl 1107.16003 · doi:10.1142/S0219498804000885
[8] L.Fuchs, ‘Cotorsion and Tor pairs and finitistic dimensions over commutative rings’, in: Groups, Modules, and Model Theory—Surveys and Recent Developments (Springer, Cham, 2017), 317-330.10.1007/978-3-319-51718-6_16 · Zbl 1436.13025 · doi:10.1007/978-3-319-51718-6_16
[9] L.Fuchs and S. B.Lee, ‘The functor Hom and cotorsion theories’, Comm. Algebra37 (2009), 923-932.10.1080/00927870802278602 · Zbl 1166.13013 · doi:10.1080/00927870802278602
[10] L.Fuchs and L.Salce, Modules over Non-Noetherian Domains, Mathematical Surveys and Monographs, 84 (American Mathematical Society, Providence, RI, 2001). · Zbl 0973.13001
[11] R.Göbel and J.Trlifaj, Approximations and Endomorphism Algebras of Modules, Expositions in Mathematics, 41 (Walter de Gruyter, Berlin, 2006).10.1515/9783110199727 · Zbl 1121.16002 · doi:10.1515/9783110199727
[12] R. M.Hamsher, ‘On the structure of a one-dimensional quotient field’, J. Algebra19 (1971), 416-425.10.1016/0021-8693(71)90099-8 · Zbl 0225.13005 · doi:10.1016/0021-8693(71)90099-8
[13] D. K.Harrison, ‘Infinite abelian groups and homological methods’, Ann. of Math. (2)69 (1959), 366-391.10.2307/1970188 · Zbl 0100.02901 · doi:10.2307/1970188
[14] S. B.Lee, ‘On divisible modules over domains’, Arch. Math.53 (1989), 259-262.10.1007/BF01277061 · Zbl 0652.13007 · doi:10.1007/BF01277061
[15] S. B.Lee, ‘Weak-injective modules’, Comm. Algebra34 (2006), 361-370.10.1080/00927870500346339 · Zbl 1094.13012 · doi:10.1080/00927870500346339
[16] S. B.Lee, ‘Modules over AW domains’, Preprint.
[17] E.Matlis, ‘Divisible modules’, Proc. Amer. Math. Soc.11 (1960), 385-391.10.1090/S0002-9939-1960-0116044-8 · Zbl 0095.02403 · doi:10.1090/S0002-9939-1960-0116044-8
[18] E.Matlis, ‘Cotorsion modules’, Mem. Amer. Math. Soc.49 (1964). · Zbl 0135.07801
[19] E.Matlis, 1-Dimensional Cohen-Macaulay Rings, Lecture Notes in Mathematics, 327 (Springer, Berlin, 1973).10.1007/BFb0061666 · Zbl 0264.13012 · doi:10.1007/BFb0061666
[20] B.Stenström, Rings of Quotients, Grundlehren der Mathematischen Wissenschaften, 217 (Springer, New York, 1975).10.1007/978-3-642-66066-5 · Zbl 0296.16001 · doi:10.1007/978-3-642-66066-5
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