## Whitehead property of modules.(English)Zbl 0616.16009

A module N over a ring R is said to have the Whitehead property (WP) if either N is injective or else Ext$${}^ 1_ R(M,N)\neq 0$$ for all non- projective R-modules M. The left nonsingular rings of cardinality less than the continuum for which all left modules have the (WP) are completely described. Given a countable, simple, von Neumann regular, non-artinian ring R, the assertion ”every countable left R-module has the (WP)” is shown to be independent of ZFC. Finally, Cohn’s solution of Artin’s skew field problem is used to construct a ring for which all cyclic left modules have the (WP) but not all finitely generated left modules do.
Reviewer: K.R.Goodearl

### MSC:

 16D80 Other classes of modules and ideals in associative algebras 16Exx Homological methods in associative algebras 03E35 Consistency and independence results 16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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### References:

 [1] F. W. Anderson, K. R. Fuller: Rings and categories of modules. Springer-Verlag, New York-Heidelberg-Berlin, 1974. · Zbl 0301.16001 [2] L. Bican T. Kepka, P. Němec: Rings, modules, and preradicals. M. Dekker Inc., New York-Basel, 1982. · Zbl 0483.16026 [3] P. M. Cohn: Skew field constructions. Cambridge University Press, Cambridge, 1977. · Zbl 0355.16009 [4] P. C. Eklof: Independence results in algebra. mimeographed lecture notes, Yale, 1976. [5] K. R. Goodearl: Von Neumann regular rings. Pitman, London-San Francisco-Melbourne, 1979. · Zbl 0411.16007 [6] A. H. Schofield: Artin’s problem for skew field extensions. Math. Proc. Camb. Phil. Soc. 97 (1985), 1-6. · Zbl 0574.16008 [7] S. Shelah: A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals. Israel J. Math. 21 (1975), 319-349. · Zbl 0369.02034 [8] S. Shelah: Whitehead groups may be not free. even assuming CH, I., Israel J. Math. 28 (1977), 193-204. · Zbl 0369.02035 [9] J. Trlifaj, T. Kepka: Structure of $$T$$-rings. in Radical Theory (Proc. Conf. Eger, 1982), Colloq. Math. Soc. Bolyai, North-Holland, Amsterdam) · Zbl 0588.16021 [10] J. Trlifaj: Ext and von Neumann regular rings. Czech. Math. J. 35 (110), (1985), 324-332. · Zbl 0577.16007
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