So we have a ring homomorphism,

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,