Let be a commutative ring with 1, and let be ideals. Recall that . Prove the following.
- is an ideal
- If then
- If , then there exist and such that .
- If and , then .
Suppose , and let . Then since and are ideals. Moreover, . By the submodule criterion, is an ideal.
For all , . So , and similarly .
Suppose for some ideal . Since is an ideal, it is closed under sums, so that for all and . Thus .
Note that , so that by the previous point. Now let ; by definition, for some and . Collecting terms in and , we have for some and . Thus .
Suppose . By the previous point, , and in particular for some and .
Suppose . By the previous point, there exist and such that . Now let . We have . Since , , and so . Thus .