$(a) = R$ if and only if a is a unit. Let $⟨R;+,−,0,·,1⟩$ be a commutative ring. For $a \in R$, define $(a)$ $:= \{a · r | r \in R \}.$ How can i prove that $(a) = R$ if and only if a is a unit. 
So if there exist  $a' \in R$ such that $a*a' = 1$ we have $ \forall r \in R $ $ (a*a')*r = r$ hence $(a) = R$ , right ?
If $(a) = R$  than  $ \forall r \in R $ there exists $r' \in R $ such that $ a*r' = r$ but than ...?
Is the first part correct and how can i formulate the second part?

The first part is correct: if $a$ is a unit, then $ab=1$ for some $b\in R$; therefore, for every $r\in R$,
$$
r=r1=r(ab)=a(rb)\in(a)
$$
hence $R\subseteq (a)$ and therefore $R=(a)$.
The second part is simpler: if $(a)=R$, then $1\in (a)$, so there exists $b\in R$ with $ab=1$.
