Equivalence of Definitions of Noetherian Ring
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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.
