Two questions.
Q1. Let $A$ be a finite-dimensional algebra over a field with block decomposition $A=A_1\oplus A_2\oplus\cdots\oplus A_r$ and $M$ an $A$-module. If every composition factor of $M$ lies in the block $A_i$ for some $i$, why must $M$ lie in $A_i$?
Q2. I would like to understand the statement of the following proposition:
Two simple $A$-modules $S$ and $T$ lie in the same block of $A$ if and only i there are simple $A$-modules $S=S_1, S_2,\ldots,S_m=T$ such that $S_i,S_{i+1}$, $1\leq i
What does this mean? We need to find a ``sequence'' of simple modules going from $S$ to $T$ such that successive modules in the sequence are composition factors of the same projective indecomposable?