Let be a commutative ring and be the ideal of . Show that is also the ideal of .
Let . Then, and for some .
Basically you need to argue the followings in order to show that is indeed an ideal of R.
1. Since , it follows immediately that .
Since , we see that . Thus .
2. .
Since and for some , we see that (Use a binomial exapansion ).
3. .
Argue that .
4. or for any .
Argue that or . Since R is a commutative ring (its ideals are two-sided), you need to show either of them.