I have just began studying the lecture notes introduction to representation theory by Etingof for several days.
I learn that every finite representation $V$ of an associative algebra $A$ admits a filtration:
$0= V_0\subset V_1\subset \cdots \subset V_n =V$ such that $V_i/V_{i-1}$ is irreducible for all positive integer $i\leq n$
From that , I encounter a confusion.
Let $W = V_1\oplus V_2/V_1\oplus \cdots \oplus V/V_{n-1}$ , then $W$ has character as $\chi_W = \chi_V+\chi_{V_2/V_1} + \cdots +\chi_{V/V_{n-1}}$ which is the same as $\chi_V$. Hence $V \cong W$
However , $V = V_1\oplus V_2/V_1\oplus \cdots \oplus V/V_{n-1}$ shows $V$ is completely reducible. In other words , every finite representation is completely reducible.
I have already known there exists finite representation which is not completely reducible , but I don't know where I make mistakes.
Please help me , thank you.