A ring $R$ is called central reduced if every nilpotent element is central. Ungor et al. math.RA 14 Dec 2013 has given an example of a commutative ring which is central reduced but not reduced. Can we find an example of a ring which is noncommutative central reduced but not reduced?
The answer is yes, and there are many ways to do it.
Use Jeremy Rickard's direct product construction.
Let $F$ be a field, let $R=F[x\ :\ x^2=0]$, and let $S=R\langle y,z\rangle$ be the extension of $R$ in the noncommuting variables $y,z$.
By the way, I'd personally use "nilpotent-central" to describe this "central reduced" condition, since I think the first phrase is more descriptive of what is happening (we are forcing all nilpotent elements to be central).