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Endomorphism rings of faithfully flat Abelian groups. (English) Zbl 0709.20031
An abelian group A is called faithfully flat if A is flat when considered as a left module over its endomorphism ring End(A) and IA\(\neq A\) for all proper right ideals I of End(A). At least this is the definition given in the preprint “Abelian groups flat over their endomorphism rings” by D. Arnold (reference [Ar1]). I can only assume it is the definition the author is using, since he gives no direct evidence on the matter. The reader is referred to the paper for undefined terms used below. The main thrust of the paper is to study relationships between a faithfully flat abelian group A and certain classes defined in terms of A. For example a class \({\mathcal T}\) of abelian groups is called A-balanced closed provided the following five conditions are satisfied.
(1) For each \(G\in {\mathcal T}\), \(G=S_ A(G)=\sum \{\phi (A)|\phi\in Hom(A,G)\}.\)
(2) \({\mathcal T}\) is closed under finite direct sums.
(3) If U is a subgroup of \(G\in {\mathcal T}\) with \(S_ A(U)=U\), then \(U\in {\mathcal T}.\)
(4) If B,C\(\in {\mathcal T}\) and \(\phi\in Hom(B,C)\), then ker \(\phi\in {\mathcal T}.\)
(5) A is projective with respect to any exact sequence of elements of \({\mathcal T}.\)
Theorem 2.5. The following are equivalent for a self-small abelian group.
(a) A is faithfully flat.
(b) There exists an A-balaced closed class \({\mathcal T}\) which contains the A-projective groups.
(c) The class of A-solvable groups is the largest A-balanced closed class which contains A.
The author shows that if R is a countable reduced torsion-free ring with identity, then the Corner construction of a group A with End(A)\(\simeq R\) produces a self-small faithfully flat A. Moreover, the Black Box methods of Corner-Göbel can be modified to produce a self-small faithfully flat A with prescribed endomorphism ring.
In the third section of the paper, the author obtains several stronger versions of Theorem 2.5 by imposing additional conditions on End(A). For example, a result proved in a previous paper by the author is that if A is self-small, then A is flat as an End(A)-module and End(A) is right hereditary if and only if the class of A-projectives is A-balanced closed.
In the fourth section, the author studies the quasi-splitting of exact sequences involving groups G with \(S_ A(G)=G\).
Reviewer: C.Vinsonhaler

MSC:
20K20 Torsion-free groups, infinite rank
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
16S50 Endomorphism rings; matrix rings
16D40 Free, projective, and flat modules and ideals in associative algebras
20K25 Direct sums, direct products, etc. for abelian groups
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[1] Albrecht, Ü.; Chain conditions in endomorphism rings; Rocky Mountain Journal of Mathematics 15 (1985); 91–106. · Zbl 0588.16024 · doi:10.1216/RMJ-1985-15-1-91
[2] Albrecht, Ü.; A note on locally A-projective groups; Pac. J. of Math. 120 (1985); 1–17. · Zbl 0583.20045 · doi:10.2140/pjm.1985.120.1
[3] Albrecht, Ü.; Baer’s Lemma and Fuchs’ Problem 84a; Trans. Amer. Math. Soc. 293 (1986); 565–582. · Zbl 0592.20058
[4] Albrecht, Ü.; Abelsche Gruppen mit A-projektiven Auflösungen; Habilitationsschrift; Universität Duisburg (1987)
[5] Albrecht, Ü.; Faithful abelian groups of infinite rank; to appear in the Proc. Amer. Math. Soc.
[6] Albrecht, Ü.; The structure of generalized rank 1 groups; to appear in the Houston J. Math. · Zbl 0674.20030
[7] Albrecht, Ü.; Abelian groups, A, such that the category of A-solvable groups is preabelian; to appear in the Proceedings of the Australian Conference on Abelian Groups, 1987. · Zbl 0691.20038
[8] Arnold, D; Abelian groups flat over their endomorphism ring; preprint.
[9] Arnold, D; Finite Rank Torsion-Free Abelian Groups and Rings; Springer Lecture Notes in Mathematics 931; Springer Verlag; Berlin, Heidelberg, New York (1982). · Zbl 0493.20034
[10] Arnold, D and Lady, L.; Endomorphism rings and direct sums of torsion-free abelian groups; Trans. Amer. Math. Soc. 211 (1975); 225–237. · Zbl 0329.20033 · doi:10.1090/S0002-9947-1975-0417314-1
[11] Arnold, D and Murley, C.; Abelian Groups, A, such that Horn (A, -) preserves direct sums of copies of A; Pac. J. of Math. 56 (1975); 7–20. · Zbl 0337.13010 · doi:10.2140/pjm.1975.56.7
[12] Corner, A.L.S.; Every countable reduced torsion-free ring is an endomorphism ring; Proc. London Math. Soc. 13 (1963); 687–710. · Zbl 0116.02403 · doi:10.1112/plms/s3-13.1.687
[13] Corner, A.L.S. and Göbel, R.; Prescribing endomorphism algebras, a unified treatment; Proc. London Math. Soc. (3) 50 (1985); 447–479. · Zbl 0562.20030 · doi:10.1112/plms/s3-50.3.447
[14] Chatters, A. and Hajavnavis, C.; Rings with Chain Conditions; Research Notes in Mathematics 44; Pitman Advanced Publishing Program; Boston, Melbourne, London (1980). · Zbl 0446.16001
[15] Dugas, M. and Göbel, R.; Every cotorsion-free ring is an endomorphism ring; Proc. London Math. Soc. 45(5) (1982); 319–336. · Zbl 0506.16022 · doi:10.1112/plms/s3-45.2.319
[16] Dugas, M. and Göbel, R.; On radicals and products; Pac. J. of Math. 118 (1985); 79–103. · Zbl 0578.20050 · doi:10.2140/pjm.1985.118.79
[17] M. Dugas, A. Mader and C. Vinsonhaler; Large E-rings exist; preprint.
[18] Fuchs, L; Infinite Abelian Groups, Vol. I/II; Academic Press; London, New York (1970/73). · Zbl 0209.05503
[19] Faticoni, C.; Every countable reduced torsion-free commutative ring is a pure subring of an E-ring; Comm. in Alg. 15(12); 2545–2564 (1987). · Zbl 0653.20056 · doi:10.1080/00927878708823552
[20] Goodearl, K.; Ring Theory; Marcel Dekker; Basel, New York (1976).
[21] Hausen, J.; Modules with the summand intersection property; preprint. · Zbl 0667.16020
[22] Huber, M. and Warfield, R.; Homomorphisms between cartesian powers of an abelian group; Abelian Group Theory, Proceedings Oberwolfach 1981; Springer Lecture Notes in Mathematics 874; Springer Verlag; Berlin, Heidelberg, New York (1981); 202–227. · Zbl 0484.20024
[23] Niedzwecki, G. P. and Reid, J.; Abelian groups finitely generated and projective over their endomorphism rings, preprint. · Zbl 0791.20063
[24] Schultz, P.; The endomorphism ring of the additive group of a ring; J. Aust. Math. Soc. 15; 60–69 (1973). · Zbl 0257.20037 · doi:10.1017/S1446788700012763
[25] Stock, J.; Über die Austauscheigenschaft von Moduln; Dissertation; München (1982).
[26] Ulmer, F.; A flatness criterion in Grothendick categories; Inv. Math. 19 (1973); 331–336. · Zbl 0257.18020 · doi:10.1007/BF01425418
[27] Ulmer, F.; Localizations of endomorphism rings and fixpoints; J. of Alg. 43; 529–551 (1976). · Zbl 0378.18009 · doi:10.1016/0021-8693(76)90125-3
[28] Vinsonhaler, C. and Wickless, W.; Locally irreducible rings; Bull. Austral. Math. Soc. 32, (1985), 129–145. · Zbl 0565.20036 · doi:10.1017/S0004972700009795
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