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From: John R. Martin on 3 Feb 2010 16:29
Let k be an algebraically closed field, and A a finite dimensional k-algebra. Let l(A) denote the number (of isomorphism classes) of simple A-modules.

1) Why is l(A) <= dim(A)? This seems like it should be easy, but I don't really see it immediately.

2) If e and b are two idempotents in A, why are the A-modules Ae and Ab isomorphic if and only if e is conjugate to b (as in e = c'bc with c an invertible element of A)? One direction is easy, but I can't see how to show e and b are conjugate just from a module isomorphism.

If it helps, both these questions are related to some basic modular representation theory (in particular the theory of blocks).

Thanks,
JR
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