Let be a commutative ring with 1 and let be ideals. Prove the following. (1) , (2) , (3) If and then , and (4) .
Let and . Certainly then by our equivalent characterization of ideal products, .
By this previous exercise, if then . Each is in , so that . So . The reverse inclusion is similar.
Suppose . Since is an ideal, each is in . Then .
Since is commutative, .