Let be a commutative ring and be the ideal of . Show that is also the ideal of .

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- Dec 16th 2009, 09:26 PMGTK X HunterIdeal of ring
Let be a commutative ring and be the ideal of . Show that is also the ideal of .

- Dec 16th 2009, 10:58 PMaliceinwonderland
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.