Equivalence of Definitions of Noetherian Ring

Theorem
Let $A$ be a commutative ring with unity.

Then the following are equivalent:


 * 1. Every ideal $I \subset A$ is finitely generated.
 * 2. $A$ satisfies the ascending chain condition on subrings
 * 3. $A$ satisfies the maximal condition on subrings.

Proof
We have 2. $\iff$ 3. by Increasing Sequence in Ordered Set Terminates iff Maximal Element.

2. $\implies$ 1.

1. $\implies$ 2.