Sign up ×  I'm reading these online notes on representation
theory, and I don't fully understand this:   Isn't
$V\oplus(\bigoplus_{i\in I}S_i)$ a direct sum by definition, so
we'd get $I=\{1,\cdots,n\}$? Can we assume the $S_i$ are
pairwise disjoint submodules of $U$ based on how the lemma is
phrased (or does that follow necessarily from them each being
simple)? Intuitively, I feel the condition should be that $W$ as
an external direct sum maps into the internal direct sum
situated within $U$ in an obvious way, and we choose $I$ maximal
so that this map has no kernel. Is this understanding correct?
share|cite|improve this question I don't understand your first
question in the last parraph above: I is maximal wrt that sum
being direct, but somehow you seem to think that the sum of the
n submodules $\,S_i\,$ is direct, which is wrong. And no, why
could we assume the $\,S_i\,$ have trivial intersection (of
course, they can't be disjoint). What they do is to take sum of
$\,S_i'\,$s that (1) is direct, but also (