On Whitehead modules. (English) Zbl 0743.16004

The authors prove consistency results with ZFC+GCH, related to Whitehead and Baer modules. Their use of forcing is essential in proofs of the following main results. For a wide class of rings \(R\) it is consistent that, for every \(R\)-module \(K\) there is a non-projective module \(M\) with \(Ext^ 1_ R(M,K)=0\); consequently, there are non-projective Whitehead \(R\)-modules. The generalization is the consistency of the statement that, for some rings \(R\), there exist Whitehead \(R\)-modules which are not the union of a continuous chain of submodules with all the quotients being small Whitehead \(R\)-modules. It is undecidable in ZFC+GCH whether there is a test module for being a Baer module.


16D40 Free, projective, and flat modules and ideals in associative algebras
16B70 Applications of logic in associative algebras
03C25 Model-theoretic forcing
03C60 Model-theoretic algebra
03E35 Consistency and independence results
Full Text: DOI


[1] Bass, H., Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc., 95, 466-488 (1960) · Zbl 0094.02201
[2] Becker, T.; Fuchs, L.; Shelah, S., Whitehead modules over domains, (Forum Math., 1 (1989)), 53-68 · Zbl 0651.20061
[3] Eklof, P. C., Homological algebra and set theory, Trans. Amer. Math. Soc., 227, 207-225 (1977) · Zbl 0355.02047
[4] Eklof, P. C., Set-Theoretic Methods in Homological Algebra and Abelian Groups (1980), Univ. of Montreal Press · Zbl 0488.03029
[5] Eklof, P. C.; Fuchs, L., Baer modules over valuation domains, Ann. Mat. Pura Appl., 150, 363-374 (1988) · Zbl 0654.13014
[6] Eklof, P. C.; Fuchs, L.; Shelah, S., Baer modules over domains, Trans. Amer. Math. Soc., 322, 547-560 (1990) · Zbl 0715.13008
[7] Eklof, P. C.; Mekler, A. H., Categoricity results for \(L_{t8}\) κ-free algebras, Ann. Pure Appl. Logic, 37, 81-99 (1988) · Zbl 0648.03021
[8] Eklof, P. C.; Mekler, A. H., Almost Free Modules: Set-Theoretic Methods (1990), North-Holland: North-Holland Amsterdam · Zbl 0718.20027
[9] Fuchs, L., Infinite Abelian Groups (1973), Academic Press: Academic Press Orlando, FL · Zbl 0253.20055
[10] Fuchs, L.; Salce, L., Modules over Valuation Domains, (Lecture Notes in Pure and Applied Math, Vol. 97 (1985), Dekker: Dekker New York) · Zbl 0668.13016
[11] Jech, T., Set Theory (1978), Academic Press: Academic Press Orlando, FL · Zbl 0419.03028
[12] Kaplansky, I., Projective modules, Ann. of Math. (2), 68, 372-377 (1958) · Zbl 0083.25802
[13] Mekler, A. H.; Shelah, S., When κ-free implies strongly κ-free, (Abelian Group Theory (1987), Gordon & Breach: Gordon & Breach New York), 137-148 · Zbl 0652.20057
[14] Mekler, A. H.; Shelah, S., Uniformization principles, J. Symbolic Logic, 54, 441-459 (1989) · Zbl 0699.03028
[15] Shelah, S., Infinite abelian groups, Whitehead problem and some constructions, Isreal J. Math., 18, 243-256 (1974) · Zbl 0318.02053
[16] Shelah, S., A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Isreal J. Math., 21, 319-340 (1975) · Zbl 0369.02034
[17] Shelah, S., Whitehead groups may not be free even assuming CH, I, Israel J. Math., 28, 193-203 (1977) · Zbl 0369.02035
[18] Shelah, S., Whitehead groups may not be free even assuming CH, II, Israel J. Math., 35, 257-285 (1980) · Zbl 0467.03049
[19] Shelah, S., Proper Forcing, (Lecture Notes in Mathematics, Vol. 940 (1982), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0495.03035
[20] Shelah, S., Diamonds uniformization, J. Symbolic Logic, 49, 1022-1033 (1984) · Zbl 0598.03044
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