Kaplansky classes. (English) Zbl 1099.13019

P. Eklof and J. Trlifaj [Bull. Lond. Math. Soc. 33, No. 1, 41–51 (2001; Zbl 1030.16004)] have shown an important role in homological algebra of the cotorsion theory introduced by L. Salce [Symp. Math. 23, 11–32 (1979; Zbl 0426.20044)].
The authors define and study a Kaplansky class of modules which behave well under the technique of Eklof and Trlifaj. General existence theorems on covers and envelopes in the Gorenstein setting are proved. In particular, if \(R\) is a noetherian ring then the class of injective Gorenstein modules is a Kaplansky class, therefore every \(R\)-module has a Gorenstein injective preenvelope. Those classes are also related to \(\mu\)-dimension of modules.


13C12 Torsion modules and ideals in commutative rings
13D30 Torsion theory for commutative rings
16D10 General module theory in associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
Full Text: EuDML


[1] M. AUSLANDER, Anneaux de Gorenstein et torsion en alègebre commutative, Séminaire d’algèbre commutative 1966/67, notes by M. Mangeney, C. Peskine and L. Szpiro, École Normale Supérieure de Jeunes Filles, Paris (1967).
[2] L. BICAN - R. EL BASHIR - E. E. ENOCHS, Modules have flat covers, to appear in Bull. London Math. Soc. Zbl1029.16002 MR1832549 · Zbl 1029.16002
[3] N. DING - J. CHEN, Coherent rings with finite self-FP-injective dimension, Comm. Algebra, 24 (9) (1996), pp. 2963-2980. Zbl0855.16001 MR1396867 · Zbl 0855.16001
[4] P. EKLOF - J. TRLIFAJ, How to make Ext vanish, to appear in J. Lond. Math. Soc. Zbl1030.16004 MR1798574 · Zbl 1030.16004
[5] E. ENOCHS, Injective and flat covers, envelopes and resolvents, Israel J. Math., 39 (3), (1981), pp. 189-209. Zbl0464.16019 MR636889 · Zbl 0464.16019
[6] E. ENOCHS - O. M. G. JENDA, Balanced Functors Applied to Modules, J. Algebra, 92 (1985), pp. 303-310. Zbl0554.18006 MR778450 · Zbl 0554.18006
[7] E. E. ENOCHS - O. M. G. JENDA - B. TORRECILLAS, Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan, 10 (1993), pp. 1-9. Zbl0794.16001 MR1248299 · Zbl 0794.16001
[8] E. E. ENOCHS - O. M. G. JENDA - J. XU, Covers and envelopes over Gorenstein rings, Tsukuba J. Math., 20 (1996), pp. 487-503. Zbl0895.16001 MR1422636 · Zbl 0895.16001
[9] E. E. ENOCHS - O. M. G. JENDA, Relative Homological Algebra, De Gruyter, Berlin - New York (2000). Zbl0952.13001 MR1753146 · Zbl 0952.13001
[10] E. E. ENOCHS - O. M. G. JENDA - L. OYONARTE, l and m-dimensions of modules, to appear in Rend. Sem. Mat. Univ. Padova, 105 (2001). Zbl1072.16011 MR1834984 · Zbl 1072.16011
[11] M. HOVEY, Cotorsion Theories, Model Category Structures and Representation Theory, preprint. · Zbl 1016.55010
[12] C. JENSEN, Les Foncteurs Dérivés de lim J et leur Applications en Théorie des Modules, Lecture Notes in Math. 254, Springer-Verlag (1972). Zbl0238.18007 MR407091 · Zbl 0238.18007
[13] I. KAPLANSKY, Projective Modules, Ann. of Math., 68 (2) (1958), pp. 372-377. Zbl0083.25802 MR100017 · Zbl 0083.25802
[14] F. MAEDA, Kontinuerliche Geometrien, Springer-Verlag, Berlin (1958). MR90579
[15] L. SALCE, Cotorsion theories for abelian groups, <Symposia Mathematica> Vol. XXIII, pp. 11-32, Academic Press, London - New York (1979). Zbl0426.20044 MR565595 · Zbl 0426.20044
[16] C. WALKER, Relative homological algebra and Abelian groups, Illinois J. Math., 10 (1966), pp. 186-209. Zbl0136.25601 MR190210 · Zbl 0136.25601
[17] J. XU, Flat covers of modules, Lecture Notes in Math. 1634, Springer-Verlag, (1996). Zbl0860.16002 MR1438789 · Zbl 0860.16002
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