A-Projective Groups of Large Cardinality
Because free abelian groups have many useful homological properties, the question arises how closely do groups of the form ⊕ A for a torsion-free group A resemble free groups. It becomes apparent that it is necessary to restrict the choices for A. In  and , it has been shown that the best way of doing this is to restrict the class of rings from which the endomorphism ring E(A) = Horn (A, A) can be chosen. In these papers, A was a torsion-free, reduced abelian group whose endomorphism ring is semi-prime, right and left Noetherian, and hereditary. Such groups will be called generalized rank 1 groups. Their important properties are given in [2, Proposition 3.2]. These can be used to show that the A-projective groups which are direct summands of groups of the form ⊕I A closely resemble free abelian groups.
KeywordsAbelian Group Direct Summand Generalize Rank Endomorphism Ring Cardinal Number
Unable to display preview. Download preview PDF.
- 1.Albrecht, U.; Endomorphism rings and A-projective torsion-free abelian groups, in Abelian Group Theory, Proceedings Honolulu 1982/83, Springer LNM 1006, New York, 1983, 209–227.Google Scholar
- 2.Albrecht, U.; A note on locally A-projective abelian groups, to appear.Google Scholar
- 3.Chase, S.; On group extensions and a problem of J. H. C. Whitehead, in Topics in Abelian Groups, Scott — Foresman, Glenview, 1963, 173–193.Google Scholar