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On *-modules generating the injectives. (English) Zbl 0783.16005
A (right \(R\)-) module \(P\) is called a \(*\)-module provided \(P\) induces a category equivalence between \(\text{Gen}(P_ R)\) (= the class of all modules generated by \(P\)) and all right \(S\)-modules cogenerated by \(P^*\). Here, \(S=\text{End}(P_ R)\) and \(P^*=\operatorname{Hom}_ R(P,Q)\) for an injective cogenerator \(Q\) of \(\text{Mod-}R\). The notion of a \(*\)-module is a common generalization of the well-known notions of a quasi- progenerator and a tilting module. Denote by \(I\) the class of all injective modules. The main result of the paper shows that each \(*\)- module such that \(\text{Gen}(P_ R)\supseteq I\) is finitely generated. This result is a precursor to more recent results of R. Colpi [Commun. Algebra 21, 1095-1102 (1993)], showing that \(*\)-modules such that \(\text{Gen}(P_ R)\supseteq I\) are exactly the generalized tilting modules, and of the author [Every \(*\)-module is finitely generated, J. Algebra (to appear)].
Reviewer: L.Bican (Praha)

MSC:
16D90 Module categories in associative algebras
16D50 Injective modules, self-injective associative rings
16S50 Endomorphism rings; matrix rings
18G05 Projectives and injectives (category-theoretic aspects)
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References:
[1] F.D. Anderson - K.R. Fuller , Rings and Categories of Modules , Springer , New York ( 1974 ). MR 417223 | Zbl 0765.16001 · Zbl 0765.16001
[2] R. Colpi , Some remarks on equivalences between categories of modules , Comm. Alg. , 18 ( 1990 ), pp. 1935 - 1951 . MR 1071082 | Zbl 0708.16002 · Zbl 0708.16002 · doi:10.1080/00927879008824002
[3] R. Colpi - C. Menini , On the structure of *-modules , J. Alg. (as appear). Zbl 0795.16005 · Zbl 0795.16005 · doi:10.1006/jabr.1993.1138
[4] G. D’Este - D. Happel , Representable equivalences are represented by tilting modules , Rend. Sem. Mat. Univ. Padova , 83 ( 1990 ), pp. 77 - 80 . Numdam | MR 1066430 | Zbl 0706.16011 · Zbl 0706.16011 · numdam:RSMUP_1990__83__77_0 · eudml:108185
[5] P.C. Eklof - A.H. Mekler , Almost Free Modules , North-Holland , Amsterdam ( 1990 ). MR 1055083 | Zbl 0718.20027 · Zbl 0718.20027
[6] P.C. Eklof - S. Shelah , On Whitehead modules , J. Alg. , 142 ( 1991 ), pp. 492 - 510 . MR 1127077 | Zbl 0743.16004 · Zbl 0743.16004 · doi:10.1016/0021-8693(91)90321-X
[7] K.R. Fuller , Density and equivalence , J. Alg. , 29 ( 1974 ), pp. 528 - 550 . MR 374192 | Zbl 0306.16020 · Zbl 0306.16020 · doi:10.1016/0021-8693(74)90088-X
[8] D. Happel - C.M. Ringel , Tilted algebras , Trans. Amer. Math. Soc. , 274 ( 1982 ), pp. 399 - 443 . MR 675063 | Zbl 0503.16024 · Zbl 0503.16024 · doi:10.2307/1999116
[9] C. Menini - A. Orsatti , Representable equivalences between categories of modules and applications , Rend. Sem. Mat. Univ. Padova , 82 ( 1989 ), pp. 203 - 231 . Numdam | MR 1049594 | Zbl 0701.16007 · Zbl 0701.16007 · numdam:RSMUP_1989__82__203_0 · eudml:108160
[10] L. Salce , Cotorsion theories for abelian groups , Symp. Math. , 23 ( 1979 ), pp. 11 - 32 . MR 565595 | Zbl 0426.20044 · Zbl 0426.20044
[11] J. Trlifaj , Associative Rings and the Whitehead Property of Modules , R. Fischer , Munich ( 1990 ). MR 1053965 | Zbl 0692.16017 · Zbl 0692.16017
[12] P. Zanardo , On *-modules over valuation rings , Rend. Sem. Mat. Univ. Padova , 83 ( 1990 ), pp. 193 - 199 . Numdam | MR 1066441 | Zbl 0716.13017 · Zbl 0716.13017 · numdam:RSMUP_1990__83__193_0 · eudml:108177
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