Another exercise from Balwant-Singh:
Show that if $A$ is local then Spec($A$) is connected in the Zariski topology.
Any hint ?
Another possible hint: a local ring $R$ have a unique maximal ideal $m$. For every proper ideal $I \lhd R$ we must have $m \in V(I)$ hence every openset $m \not \in D(I)$, for $I$ proper ideal.