Given any ring R AND a subset X of R, we can consider the set of all subrings of R containing X and form their intersection, T, say.
Show that T is the smallest subring of R containing X.
This is actually a straightforward manipulation of definitions. You'll need to prove two things: is a subring, then is the smallest subring containing . Suppose your set of all subrings of containing is given by .
For the first one, you really just need to check that . The notation is a bit cumbersome, but this fact is actually quite trivial if you think about it. We also have by definition of .
The second bit is even easier: Let be a subring of containing . Since this is a subring of containing , we have by definition of . Using the basic property of intersections, this says that