Mathematical Research Letters

Volume 6, Issue 4, July 1999 pp. 371-375.

Hereditary Properties of Direct Summands of Algebras

Authors: Hanspeter Kraft, Lance W. Small, Nolan R. Wallach
Author institution: Universität Basel, and University of California, San Diego

Summary: We consider subrings $S$ of rings $R$ such that $R=S\oplus V$ with $V$ either two sided invariant under multiplication by $S$ or invariant under the commutator with $S$. We show that some important properties of $R$ are inherited by $S$ under such conditions. One is the {\rm FCR\/}-property which says that every finite dimensional representation is completely reducible. Another application gives a characterization (in characteristic zero) of reductive subgroups of reductive groups.

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