Show that $R$ is a ring Let $R$ a set with $+$ and $\cdot $ s.t.
1) $(R,+)$ is a group (not necessarily comutative)
2) $\cdot $ is associative, and distributive for $+$, i.e. $a\cdot (b\cdot c)=(a\cdot b)\cdot c$, $a\cdot (b+c)=a\cdot b+a\cdot c$ and $(a+b)\cdot c=a\cdot c+b\cdot c$
3) there is $1\in R$ s.t. $1\cdot x=x\cdot 1=x$.
Show that $R$ is a ring.
I really don't know how to did it. I think that I have to show that $(R,+)$ is commutative, but I didn't success.

Hint: 
Compute
$$
(1 + 1) \cdot (a + b)
$$
in two ways, using the two distributive laws.
