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A Note On Clean Abelian Groups, Brendan Goldsmith, P. Vamos
A Note On Clean Abelian Groups, Brendan Goldsmith, P. Vamos
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Nicholson defined a ring to be clean if every element is the sum of a unit and an idempotent. A module is clean if its endomorphism algebra is clean. We show that torsion-complete Abelian p-groups are clean and characterize the clean groups among the class of totally projective p-groups. An example is given of a clean p-group which is neither totally projective nor torsion- complete
On Cosmall Abelian Groups, Brendan Goldsmith, O. Kolman
On Cosmall Abelian Groups, Brendan Goldsmith, O. Kolman
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It is a well-known homological fact that every Abelian groupGhas the property that Hom(G,−)com-mutes with direct products. Here we investigate the ‘dual’ property: an Abelian groupGis said to be cosmallif Hom(−,G)commutes with direct products. We show that cosmall groups are cotorsion-free and that nogroup of cardinality less than a strongly compact cardinal can be cosmall. In particular, if there is a properclass of strongly compact cardinals, then there are no cosmall group