Prove that every finite ring is artinian.
Solution:
A ring A which is either finite or a finite dimensional algebra over a field. It is clear that such a ring satisfies the descending chain condition for ideals. That is to say any descending chain of ideals I0 I1 is constant after some i. A ring satisfying this condition is also called an Artinian ring in honour of Emil Artin.
Further, a ring R is called Artinian if it satisfies the defending chain condition on ideals.
That is, whenever we have ideals of R satisfying
there is an integer N such that
Comments
Leave a comment