Problem with structure of a semisimple ring theorem

Question

Structure of semisimple ring (Wedderburn-Artin) in Rings and Categories of Modules - Frank W. Anderson, Kent R. Fuller (auth.)

Proof:

Please explain that:

1. "Now $_RR$ is direct sum off these traces". It's OK!
2. "so (see 7) there is a finite set...". I read unit 7 and can't found that.
3. $T_i$ is simple left ideal of the ring $Tr_R(T_i)$
4. "Thus by 13.5 ..."

Thanks!

Answer 1

So, turning back to 9.12, you see

9.12 Proposition. The socle of a left $R$-module $M$ is, as a left $R$-right $End(_RM)$-bimodule, a direct sum of its homogenous components.

In the discussion before this proposition, they give you the information to see that $Tr_R(T_i)$ is the homogenous component of $T_i$ in $M$. Explicitly, $Tr_R(T_i)=\sum \{_RS<_RR\mid _RS\cong _RT_i\}$.

For the case you are working on, $_RR=_RM$. Since $soc(_RR)=R$, Proposition 9.12 says for you "$_RR$ is a direct sum of $Tr_R(T_i)$".