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

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Robert · Accepted Answer

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,

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

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