# zbMATH — the first resource for mathematics

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.

##### MSC:
 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:
##### References:
 [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, (), 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 Lt8 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 Amsterdam · Zbl 0718.20027 [9] Fuchs, L, Infinite abelian groups, (1973), Academic Press Orlando, FL · Zbl 0253.20055 [10] Fuchs, L; Salce, L, Modules over valuation domains, () · Zbl 0668.13016 [11] Jech, T, Set theory, (1978), 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, (), 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, () · Zbl 0495.03035 [20] Shelah, S, Diamonds uniformization, J. symbolic logic, 49, 1022-1033, (1984) · Zbl 0598.03044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.