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.: Assume there is an ideal $I$ which is not finitely generated. For any finite set $\{ a_1, ..., a_n \}$, $n \in \mathbf{N}$, the generated ideal is not equal to $I$. Consider the chain $\langle a_1 \rangle \subset \langle a_1, a_2 \rangle \subset \cdots$. This chain does not satisfy the ascending chain condition (note that $I$ has infinitely many elements by assumption).

1. $\implies$ 2.: Let there be a chain of ideals $I_1 \subset I_2 \subset \cdots$. Then $J = \bigcup_{n \geq 1} I_n$ is an ideal. As $J$ is finitely generated, say by $\{ b_1, ..., b_m \}, m \in \mathbf{N}$, we can find an ideal such that $\{ b_1, ..., b_m \} \subset I_k$ for some $k \in \mathbf{N}$ (since the chain is ascending). Clearly $I_k = \langle b_1, ..., b_m \rangle$ and hence for all $l \geq k$: $I_l = I_k$.