The Kaplansky test problems – An approach via radicals. (English) Zbl 0847.20051
Assuming Jensen’s diamond principle, a new construction is given of strongly \(\lambda\)-free Abelian groups and modules with a given endomorphism algebra. This implies negative answers to the Kaplansky Test Problems for strongly \(\aleph_1\)-free Abelian groups of cardinality \(\aleph_1\) assuming the diamond principle; these and stronger results were already proved by the first author and M. Dugas [Proc. Lond. Math. Soc., III. Ser. 45, 319-336 (1982; Zbl 0506.16022)], but the new approach here may be useful. It should be noted that no results about the Kaplansky Test Problems are proved here in ZFC despite an inadvertent apparent assertion to the contrary in the Abstract.
Reviewer: P.Eklof (Irvine)
MSC:
| 20K20 | Torsion-free groups, infinite rank |
| 03E35 | Consistency and independence results |
| 20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |
| 16S50 | Endomorphism rings; matrix rings |
Keywords:
diamond principle; strongly \(\lambda\)-free Abelian groups; endomorphism algebras; Kaplansky’s test problems; strongly \(\aleph_ 1\)-free Abelian groupsCitations:
Zbl 0506.16022References:
| [1] | 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 |
| [2] | Corner, A. L.S., On a conjecture of Pierce concerning direct decompositions of Abelian groups, (Proceedings of the Colloquium on Abelian Groups. Proceedings of the Colloquium on Abelian Groups, Tihany, Budapest (1964)), 43-48 · Zbl 0132.27201 |
| [3] | Corner, A. L.S., On endomorphism rings of primary Abelian groups, Quart. J. Math. Oxford, 20, 277-296 (1969) · Zbl 0205.32506 |
| [4] | Corner, A. L.S., Every countable reduced torsion-free algebra is an endomorphism algebra (1972), unpublished manuscript, ca · Zbl 0116.02403 |
| [5] | Corner, A. L.S.; Göbel, R., Prescribing endomorphism algebras, a unified treatment, (Proc. London Math. Soc. (3), 50 (1985)), 447-479 · Zbl 0562.20030 |
| [6] | Dugas, M.; Göbel, R., Every cotorsion-free ring is an endomorphism ring, (Proc. London Math. Soc. (3), 45 (1982)), 319-336 · Zbl 0506.16022 |
| [7] | Dugas, M.; Göbel, R., On radicals and products, Pacific J. Math., 118, 79-104 (1985) · Zbl 0578.20050 |
| [8] | K. Eda, Cardinality restrictions on preradicals, in: Abelian Group Theory, Proceedings of the 1987 Perth Conference, Contemporary Mathematics, Vol. 87, (American Mathematical Society, Providence, RI) 277-283.; K. Eda, Cardinality restrictions on preradicals, in: Abelian Group Theory, Proceedings of the 1987 Perth Conference, Contemporary Mathematics, Vol. 87, (American Mathematical Society, Providence, RI) 277-283. · Zbl 0687.20049 |
| [9] | Eklof, P. C., On the existence of κ-free Abelian groups, (Proc. Amer. Math. Soc., 47 (1975)), 65-72 · Zbl 0268.20036 |
| [10] | Eklof, P. C.; Mekler, A. H., Almost Free Modules, Set-Theoretic Methods (1990), North-Holland: North-Holland Amsterdam · Zbl 0718.20027 |
| [11] | Fuchs, L., (Infinite Abelian Groups, Vols. I and II (1973), Academic Press: Academic Press New York) · Zbl 0257.20035 |
| [12] | Göbel, R., An easy topological construction for realizing endomorphism rings, (Proc. Royal Irish Acad. Sect., A 92 (1992)), 281-284 · Zbl 0806.16031 |
| [13] | Göbel, R.; Goldsmith, B., Cotorsion-free algebras as endomorphism algebras in \(L\) — The discrete and the topological case, Comment. Math. Univ. Carolin, 34, 1, 1-9 (1993) · Zbl 0804.16031 |
| [14] | Göbel, R.; May, W., Independence in completions and endomorphism algebras, (Forum Math., 1 (1989)), 215-226 · Zbl 0691.13004 |
| [15] | Griffith, P., \(ℵn- free\) Abelian groups, Quart. J. Math. Oxford, 23, 2, 417-425 (1972) · Zbl 0274.20068 |
| [16] | Hill, P., New criteria for freeness in Abelian groups, II, Trans. Amer. Math. Soc., 196, 191-201 (1974) · Zbl 0296.20026 |
| [17] | Jech, T., Set Theory (1978), Academic Press: Academic Press New York · Zbl 0419.03028 |
| [18] | Kaplansky, I., Infinite Abelian Groups (1969), University of Michigan Press: University of Michigan Press Ann Arbor, MI · Zbl 0194.04402 |
| [19] | M. Magidor and S. Shelah, Construction of almost-free groups in ZFC, Israel J. Math., to appear.; M. Magidor and S. Shelah, Construction of almost-free groups in ZFC, Israel J. Math., to appear. · Zbl 0819.20059 |
| [20] | Shelah, S., A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel J. Math., 21, 319-349 (1975) · Zbl 0369.02034 |
| [21] | Thomé, B., \(ℵ1- separable\) groups and Kaplansky’s test problems, (Forum Math., 2 (1990)), 203-212 · Zbl 0694.20035 |
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