Show a module is isomorphic to a direct sum of $\alpha N$, where $N$ is a minimal ideal I am studing the book "Associative Algebra" written by Richard S. Pierce. On page 45,  I am at a loss for the Proposition a and b, and Proposition 2.5 to solve the Corollary a.
Corollary a. Let A be a simple algebra,and suppose that $N$ is a minimal right ideal of $A$. If $M$ is a right A-module, then there is a unique cardinal number $\alpha$ such that $M\cong \bigoplus \alpha N$. 
I list Proposition a and b, and Proposition 2.5 in the following picter. The third Proposition is Proposition 2.5. 


 A: Obviously if you have a simple algebra $A$ with minimal right ideal $N$ as given, Proposition a) says that $A$ is semisimple.
If your definition of a semisimple ring is "every right $A$-module is a direct sum of simple $A$-modules", you now know $M\cong \bigoplus_{i\in \alpha} N_i$ where $N_i$ are simple, but potentially nonisomorphic right $A$-modules. 
But then you apply proposition b) and say that they are all mutually isomorphic to the minimal right ideal, so $M\cong\bigoplus_{i\in \alpha}N$.
The last proposition guarantees uniqueness of $\alpha$.

Proposition: $A$ is a semisimple right $A$-module iff every right $A$-module is semisimple.
$\impliedby$ is clear.
For $\implies$, let $M$ be any right $A$-module. There exists some minimal cardinal $\beta$ such that there is an surjective homomorphism $f:\bigoplus_{i\in \beta}A\to M$. Since $A$ is a direct sum of simple submodules, $\bigoplus_{i\in \beta}A$ is a direct sum of simple submodules. By the first isomorphism theorem, and the fact that quotients of semisimple modules are semisimple, the quotient $\bigoplus_{i\in \beta}A/\ker(f)\cong M$ is semisimple too.
