Recall that if is a ring with ideals and , the ideal quotient in is defined to be . This set is certainly an ideal of .
- Suppose is an integral domain, is in , and an ideal. Prove that if is a generating set for where , then is a generating set for .
- Let be a commutative ring, an ideal, and with . Prove that .
- Suppose . Then , so . Certainly , so that . Say . Since is an integral domain, , and thus . Conversely, suppose , with . Then , and thus . So , and thus . Hence .
- Suppose . Then , and in particular for all . So for all , and we have . Conversely, suppose . Then for all , and for all . Now , so that . Thus .