In general, if are ideals of such that is contained in , is contained in , and so on, then there is some for which all . That is, after some point all the ideals are equal to each other. Therefore, the ideals of satisfy the ascending chain condition, where ideals are ordered by set inclusion. Hence is a Noetherian ring.
In general, if are ideals, of such that is contained in  is contained in , and so on, then there is some for which all . That is, after some point all the ideals are equal to each other. Therefore, the ideals of satisfy the ascending chain condition, where ideals are ordered by set inclusion. Hence is a Noetherian ring.