Universal co-extensions of torsion abelian groups.

In [16] , a theory of universal extensions in abelian categories is developed; in particular, the notion of Ext 1 -universal object is presented. In the present paper, we show that an Ab3 abelian category which is Ext 1 -small satisfies the Ab4 condition if, and only if, each one of its objects is Ext 1 -universal. We use the dual of this result to construct projective effacements in Grothendieck categories. In particular, we complete the classical result of Roos on Grothendieck categories which are Ab4*, with a new proof, independent of [20]. We also give a characterization of the co- Ext 1 -universal objects of the category of torsion abelian groups. In particular, we show that such groups are the ones admitting a decomposition Q ⊕ R , in which Q is injective and R is a reduced group in which each p -component is bounded. [ABSTRACT FROM AUTHOR]