I am uncertain about the definition of irreducible components of a scheme:
In Liu, Q's algebraic geometry, it is defined as an irreducible closed subscheme which is maximal over all irreducible closed subschemes. So for example on $Spec\,A[x,y]$, $\overline{(x)/(xy)}$ and $\overline{(y)/(xy)}$ and they have intersection which is the maximal ideal $((x)/(xy),\,(y)/(xy))$.
And in other contexts, it is defined as the maximal irreducible subspace of its underlying topological space.
My question is, once we find an irreducible component in the second topological setting on its underlying space, is the corresponding irreducible closed subscheme on it as in the first definition just the reduced induced closed subscheme?
Thank you so much!!