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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Paul F. Dubois
Journal: Trans. Amer. Math. Soc. 159 (1971), 245-255
MSC: Primary 20.30
DOI: https://doi.org/10.1090/S0002-9947-1971-0280585-5
MathSciNet review: 0280585
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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 β€œaccessible". Also, these groups are the ${p^\alpha }$-high injectives.


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Keywords: Pure-injective, <!– MATH ${p^\alpha }$ –> <IMG WIDTH="27" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${p^\alpha }$">-complete, injectively closed class, high injective
Article copyright: © Copyright 1971 American Mathematical Society