Proof
Let S denote the intersection of which we must prove that it is a subring of R.
We verify the conditions for S to be a subring.
0, 1 
S
S is closed under * and +
Since each H C is a subring, we
have 0, 1 H. Hence 0, 1 belong to the
intersection over all H, that is, to
S.
C is a subring, we
have 0, 1 H. Hence 0, 1 belong to the
intersection over all H, that is, to
S.
Suppose a, b S.
Then, for each H S,
we have a, b H,
whence (as H is a subring) a + b H.
It follows that a + b S.
S.
Then, for each H S,
we have a, b H,
whence (as H is a subring) a + b H.
It follows that a + b S.
The proof for the operation * is very similar.
Compare this with the corresponding results for monoids and groups.