prove that an ideal of a ring R is a subring iff I=0 or I=R.

Question: Prove that an ideal I of a ring R is a subring if and only if I=0 or I=R

Proof:
An ideal ‘I’ is defined as,
 0 must be a part of ‘I’.
The criterion for a subring
A non-empty subset S of R is a subring if .
Both, I=0 and I=R, satisfy the criterion for an ideal. Hence I=0 and I=R are ideals.
If I=0:
Then it satisfies the condition that . Since 0 is the only element of the
set.
If I=R:
It satisfies the above condition for a subring because it itself is a ring and we know that a ring is
subring to itself.
Hence if I=0 or I=R, The ideal I...