The hypercenter of an algebraic group

We show that any connected algebraic group $G$ over a field admits a nilpotent normal subgroup $Z_\infty(G)$ such that the quotient $G/Z_\infty(G)$ has trivial center. We construct $Z_\infty(G)$ as the final term of the transfinitely extended upper central series of $G$; accordingly, we call it the hypercenter of $G$. We establish several related results about the upper central series of $G$, along with an analogue for algebraic groups of a well-known theorem of Fitting's.