And is an ideal in .
Thus, is an additive subgroup of . From group theory you should be know that the inverse image in a group homomorphism preserves subgroups. Thus, is an additive subgroup of .
All we need to show is that,
I will show that,
and leave the second anagolus part to thee to prove.
Let thus, where
Now, (by homomorphism)
and and but because is an ideal! Thus,