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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The sum-product splitting property and injective direct sums of modules over von Neumann regular rings

Author: Birge Zimmermann-Huisgen
Journal: Proc. Amer. Math. Soc. 83 (1981), 251-254
MSC: Primary 16A52; Secondary 16A30
MathSciNet review: 624908
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Abstract: Let $ {({M_i})_{i \in I}}$ be a family of modules over a von Neumann regular ring. It is shown that for the splitness of the canonical inclusion $ { \oplus _{i \in I}}{M_i} \subset \prod\nolimits_{i \in I} {{M_i}} $ it is necessary and sufficient that there be a finite subset $ I'$ of $ I$ such that the restricted sum $ { \oplus _{i \in I\backslash I'}}{M_i}$ is semisimple with finitely many homogeneous components, all simple summands being finite dimensional over their endomorphism rings. This yields a characterization of those families of injectives whose direct sum is again injective.

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Article copyright: © Copyright 1981 American Mathematical Society