Let $R$ a prime ring and $a,b\in R$ such that $ab,a\in Z(R),$ center of the ring $R$. Can we say that $b\in Z(R)?$
A ring $R$ is said to be prime if $aRb=0\implies $ either $a=0$ or $b=0$.
If the ring $R$ has unity and $a^{-1}$ exists then clearly $b\in Z(R)$, but in case when the ring $R$ need not have unity, then what can we say about $b$?
To root the conclusion in the definitions, assuming $a\neq 0$:
Consider an arbitrary $c\in R$. Then
$aR(bc-cb)=R(abc-cba)=R(abc-cab)=R(abc-abc)=\{0\}$
where the first two equalities use the centrality of $a$ and the third one uses centrality of $ab$.
By the definition of a prime ring, $bc-cb=0$. Since $c$ was arbitrary, we have found that $b$ is central.
If $a=0$, then $b$ can be anything and $ab$ is central too, so there's no way to conclude $b$ is central.