What is the scheme structure we give for closed irreducible subschemes? (or for irreducible components)

Question

In many books I have seen that the function field is defined for integral schemes. However, many times they refer in further chapters for a function field of the irreducible component of scheme (not necessarily reduced). For example, here in the answer we have only the assumption that the variety is irreducible, while in Liu's book a variety is not assumed to be reduced. So, my question is: do we always assume in such cases that we induce the reduce structure on the irreducible components? Or in closed irreducible subsets in general?