Answer to Question #201773 in Abstract Algebra for Komal

Question

Prove that every finite ring is artinian.

Answer 1

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 $I_{n}$ of R satisfying

$I_{1} \supset I_{2} \supset \cdots \supset I_{n} \supset \cdots$

there is an integer N such that

$I_{N}=I_{N+1}=I_{N+2}=\cdots$