Equivalence of Definitions of Noetherian Ring
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Theorem
The following definitions of the concept of Noetherian Ring are equivalent:
Definition 1
A commutative ring with unity $A$ is Noetherian if and only if every ideal of $A$ is finitely generated.
Definition 2
A commutative ring with unity $A$ is Noetherian if and only if it satisfies the ascending chain condition on ideals.
Definition 3
A commutative ring with unity $A$ is Noetherian if and only if it satisfies the maximal condition on ideals.
Definition 4
A commutative ring with unity $A$ is Noetherian if and only if it is Noetherian as an $A$-module.
Proof
Definition 2 iff Definition 3
This follows by Increasing Sequence in Ordered Set Terminates iff Maximal Element.
$\Box$
Definition 2 implies Definition 1
Assume there is an ideal $I$ which is not finitely generated.
For any finite set $\set {a_1, \dotsc, a_n}$ where $n \in \N$, the generated ideal is not equal to $I$.
Consider the chain:
- $\sequence {a_1} \subset \sequence {a_1, a_2} \subset \cdots$
This chain does not satisfy the ascending chain condition (note that $I$ has infinitely many elements by assumption).
$\Box$
Definition 1 implies Definition 2
Let there be a chain of ideals $I_1 \subset I_2 \subset \cdots$.
Then $\ds J = \bigcup_{n \mathop \ge 1} I_n$ is an ideal.
Let $J$ be finitely generated, by $\set {b_1, \dotsc b_m}$ for some $m \in \N$.
As the chain is ascending, there exists an ideal such that:
- $\set {b_1, \dotsc b_m} \subset I_k$
for some $k \in \N$.
It follows that:
- $I_k = \ideal {b_1, \dotsc, b_m}$
Hence:
- $\forall l \ge k: I_l = I_k$
$\Box$
Definition 1 iff Definition 4
Let $A$ is a ring.
For any subset $B\subseteq A$, we have:
- $B$ is an ideal if and only if $B$ is a submodule.
- $B$ is a finitely generated ideal if and only if $B$ is a finitely generated module.
The claim follows from these observations.
$\Box$
 | This theorem requires a proof. In particular: another proof using Definition:Noetherian Module and definition 4 You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |