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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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