# General Ring Theory/Noetherian rings

Definition (right-noetherian ring):

A commutative ring ${\displaystyle R}$ is called right-noetherian iff the set of all right ideals of ${\displaystyle R}$, ordered by inclusion, satisfies the ascending chain condition.

Left-noetherian rings are similarly defined.

Definition (noetherian ring):

A commutative ring ${\displaystyle R}$ is called noetherian iff the set of all ideals of ${\displaystyle R}$, ordered by inclusion, satisfies the ascending chain condition.

We will state and prove results only for right-noetherian rings, even though they are valid mutatis mutandis for left-noetherian and noetherian rings just as well.

Proposition (ideals of noetherian rings contain powers of their radicals):

Let ${\displaystyle R}$ be a noetherian ring, and let ${\displaystyle I\leq R}$ be an ideal. Then there exists ${\displaystyle n\in \mathbb {N} }$ such that

${\displaystyle {\sqrt {I}}^{n}\subseteq I}$.

Proof: Let ${\displaystyle i_{1},\ldots ,i_{k}}$ be a basis of ${\displaystyle sqrt{I}}$ considered as an ${\displaystyle R}$-module. Then choose ${\displaystyle m\in \mathbb {N} }$ sufficiently large so that

${\displaystyle \forall j\in [k]:i_{j}^{m}\in I}$.

Then define ${\displaystyle n:=km}$ and observe that whenever ${\displaystyle a=r_{1}i_{1}+\cdots +r_{k}i_{k}\in {\sqrt {I}}}$, then by the pigeonhole principle, upon expanding the expression ${\displaystyle a^{n}}$ and considering each summand, we find that for each summand there is at least one ${\displaystyle j\in [k]}$ so that the corresponding power of ${\displaystyle i_{j}}$ is bigger than or equal to ${\displaystyle m}$. ${\displaystyle \Box }$

Proposition (elements of noetherian rings are products of irreducible elements):

Let ${\displaystyle R}$ be a noetherian ring, and let ${\displaystyle a\in R}$ be a non-unit. Then there exist irreducible elements ${\displaystyle b_{1},\ldots ,b_{n}}$ so that

${\displaystyle a=b_{1}\cdots b_{n}}$.
(On the condition of the dependent choice.)

Proof: Indeed, a non-unit ${\displaystyle a}$ factors as ${\displaystyle a=c_{1}c_{2}}$, where ${\displaystyle c_{2}}$ is a non-unit and either ${\displaystyle c_{2}}$ is irreducible and ${\displaystyle c_{1}}$ is a unit, or ${\displaystyle c_{1},c_{2}}$ are both irreducible, or ${\displaystyle c_{2}}$ is not irreducible and a proper divisor of ${\displaystyle a}$. The same is true for ${\displaystyle c_{2}=c_{3}\cdot c_{4}}$, and proceeding inductively we gain an ascending chain

${\displaystyle \langle a\rangle \subseteq \langle c_{2}\rangle \subseteq \langle c_{4}\rangle \subseteq \langle c_{6}\rangle \subseteq \cdots }$

which stabilizes by the noetherian assumption. But if ${\displaystyle n\in \mathbb {N} }$ is chosen large enough so that the sequence stabilizes after ${\displaystyle c_{2n}}$, ${\displaystyle c_{2n}}$ is irreducible. Hence, we may factor ${\displaystyle a=d_{1}b_{1}}$, where ${\displaystyle b_{1}}$ is irreducible, and continuing in the same fashion we obtain again an ascending sequence, whose stabilization implies the desired factorisation. ${\displaystyle \Box }$