Generally $p^{\alpha }$-torsion complete abelian groups

Abstract:

A generalized $p$-primary cotorsion abelian group $G$ is a ${p^\alpha }$-injective, that is satisfies ${p^\alpha }\operatorname {Ext} ( - ,G) = 0$, iff ${G_t}$ is ${p^\alpha }$-injective in the category of torsion abelian groups. Such a torsion group is generally ${p^\alpha }$-torsion complete, but an example shows that all its Ulm factors need not be complete. The injective properties of generally ${p^\alpha }$-torsion complete groups are investigated. They are an injectively closed class, and the corresponding class of sequences is the class of ${p^\alpha }$-pure sequences with split com-c pletion when $\alpha$ is “accessible". Also, these groups are the ${p^\alpha }$-high injectives.