Mathematical Research Letters

Volume 6 (1999)

Number 4

Hereditary Properties of Direct Summands of Algebras

Pages: 371 – 375



Hanspeter Kraft

Lance W. Small

Nolan R. Wallach


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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