Associative rings and the Whitehead property of modules.

*(English)*Zbl 0692.16017
Algebra Berichte, 63. München: Verlag Reinhard Fischer. vi, 40 p. (1990).

This monograph builds on the concept (suggested by the well known Whitehead problem in abelian group theory) of a module with the Whitehead property, defined as a module N which is either injective or has the property that \(Ext(M,N)=0\) implies M projective. The author is interested in the rings R such that each R-module has the Whitehead property. These rings arise naturally in a different way when one considers the so called orthogonal theories, i.e., pairs of classes of modules (A,B) such that there exists a bifunctor F from the category of R-modules to the category of abelian groups satisfying \(A=\{X|\) \(F(X,Y)=0\) for each Y of \(B\}\) and \(B=\{Y|\) \(F(X,Y)=0\) for each X of \(A\}\). The orthogonal theories defined by F form a lattice \(L_ F\) and R is called an F-ring when \(L_ F\) has at most two elements. The author studies F-rings for several familiar bifunctors F and, in particular, Ext-rings, which are precisely the rings such that each R-module has the Whitehead property (these rings were first considered by L. Bican, T. Kepka and P. Němec [Rings, modules and preradicals (1982; Zbl 0483.16026)]). Several classes of Ext-rings are investigated in the monograph and, in particular, a structure theorem is obtained for the class of nonsingular artinian Ext-rings, which are shown to be semi-simple rings “plus” rings which are Morita equivalent to upper triangular matrix rings of degree two over skew fields. The author also considers some classes of rings which are close to being Ext-rings in the sense that certain classes of modules satisfying a determined finiteness or countability condition have the Whitehead property. For instance, he considers simple countable von Neumann regular rings which are not semisimple and looks at the problem of when the countable modules over these rings have the Whitehead property. Here, as in the classical Whitehead problem, set theory enters the picture, for it is shown that the answer to this problem is independent of \(ZFC+GCH\) (Zermelenko-Fraenkel set theory with the Axiom of Choice and the Generalized Continuum Hypothesis).

Reviewer: J.L.Gómez Pardo

##### MSC:

16D80 | Other classes of modules and ideals in associative algebras |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16Exx | Homological methods in associative algebras |

16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |

16S50 | Endomorphism rings; matrix rings |

16B50 | Category-theoretic methods and results in associative algebras (except as in 16D90) |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

16D50 | Injective modules, self-injective associative rings |