"Unique simple object" of category of modules, simplicity and irreducible modules I can't seem to find a decent definition of "unique irreducible module" for an algebra. In what sense is the uniqueness? 
For example, does $\mathfrak{sl}_2$ have a unique irreducible module, since it has one for any dimension? Or not?
Also, I am interested in the relation between those three concepts.
1) $A$ is a simple algebra
2) $A$ has a unique irreducible module.
3) The category of finitely generated $A$ modules has a unique simple object

Thanks to anyone who will help me
 A: "Unique irreducible module" means that there is a unique isomorphism class of irreducible modules; that is, every two irreducible modules are isomorphic. $\mathfrak{sl}_2$ does not have a unique irreducible module, since it has countably many nonisomorphic irreducible modules. 
A: You need to be a bit more specific about what you mean by an algebra.
If you are in the setting of finite dimensional associative algebras then the Artin-Wedderburn theorem tells us that any such simple algebra is a matrix ring $M_n(D)$ of a finite dimensional division algebra $D$ over the ground field $k$. In this case there is a unique irreducible module, namely $D^n$ (which is the unique simple in the category of $A$ modules).
Note that the converse is not quite true in this setting, the algebra $k[x]/(x^2)$ has a unique simple module $k$ where $x$ acts by zero. However it is not a simple algebra since it has the ideal generated by $x$, and its category of modules is not semisimple.
If you are talking about Lie algebras, then your example of $\mathfrak{sl}_2$ shows that a simple Lie algebra does not have a unique irreducible representation. Note that the universal enveloping algebra of $\mathfrak{sl}_2$ is neither finite dimensional nor simple so this doesn't violate the above discussion.
One more case to consider is that of an infinite dimensional simple associative algebra.  One of the simplest such examples is the Weyl algebra $\mathbb{C}[x,y]/(xy-yx-1)$, which has no finite dimensional modules and infinitely many non-isomorphic irreducible infinite dimensional modules. 
