Let $R$ be a ring with a center $Z=Z(R)$ and $S\leq R$ a sub-ring. prove that $R=ZS$ if and only if there is some subset $Z_0\subseteq Z$ such that $R=S[Z_0]$
On side I proved with no problem:
$$R=ZS\Rightarrow R =ZS\subseteq Z\cap S\Rightarrow R\subseteq Z\cup S \Rightarrow R\subseteq S[Z] \Rightarrow R=S[Z]$$
So we just pick $Z_0:=Z$
I can't prove the second direction, I would appreciate any answer.
$Z_0\subseteq Z$ so $R=S[Z_0]\subseteq S[Z]=SZ$. But $SZ\subseteq R$ by definitions of $S$ and $Z$. Thus $R=SZ$.