# Tuesday, August 23

## Noetherian and Artinian Rings

:::{.remark}
Continuing our recollection of commutative algebra:

- Maximal ideals are maximal among *proper* ideals.
  By Zorn's lemma, every proper ideal is contained in some maximal ideals.
- $I$ is maximal iff $R/I$ is a field
- $I$ is prime iff $xy\in I\implies x\in I$ or $y\in I$.
- $I$ is prime iff $R/I$ is a domain (nonzero and has no nonzero zero divisors).
- Fields are domains, so maximals are prime.
- $\spec R$ is the set of prime ideals, and $\mspec R$ maximal ideals.
  These sets carry the Zariski topology, which we will likely not use.
- The only notion of dimension we'll use is **Krull dimension**: the $\krulldim R \in \ZZ_{\geq 0}$ (if it exists) such that the length (i.e. number of *links*) of any chain of primes is bounded above by $d$:
\[
\mfp_0 \subsetneq \mfp_1 \subsetneq \cdots \subsetneq \mfp_{\ell},\qquad\ell \leq d
.\]

:::

:::{.exercise title="?"}
Show that 

- Artinian rings have dimension 0,
- Fields have dimension 0,
- Finite rings have dimension 0,
- $\krulldim \ZZ = 1$.
- If $R$ is a PID (by convention, not a field), $\krulldim R = 1$.
- Dedekind domains have dimension 1.

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:::{.remark}
Chain conditions:

- $(X, \leq)$ a poset satisfies the ACC if $\ZZ_{>0} \not\injects X$ (order-embedding), i.e. there is no strictly increasing unbounded chain $x_1 < x_2 < \cdots$.
- Posets satisfying ACC are **Noetherian**.
- $(X, \leq)$ satisfies the DCC if $\ZZ_{<0} \not\injects X$, so no chains $x_1 > x_2 > \cdots$.  
- Posets satisfying DCC are **Artinian**.
- Define order duals $X\dual = (X, \leq\dual)$ where $x\leq\dual y \iff y\leq x$, then $X$ is Noetherian iff $X\dual$ is Artinian.
  This kind of symmetry will be broken later though.
- General philosophy: generalize studying ideals in $R$, study $R$ by studying $\rmod$.
  Given a notion that makes sense for ideals, does it make sense for modules?
- For $M\in \rmod$, define $\Sub_R(M)$ to be the poset of submodules of $M$ partially ordered by inclusion.
- $M$ is Noetherian (resp. Artinian) if $\Sub_R(M)$ is Noetherian (resp. Artinian).
- Finite modules literally mean that the underlying set has finite cardinality.
- $R$ is Noetherian (resp. Artinian) as a ring iff $R$ is Noetherian (resp. Artinian) as an $R\dash$module.

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:::{.exercise title="?"}
$(X, \leq)$ satisfies ACC if every nonempty subset has a maximal element.
State and prove an analogue of this for the DCC.
:::

:::{.exercise title="?"}
Show that $M$ is Noetherian iff every submodule is finitely generated.
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:::{.warnings}
There is no analogue of this for Artinian modules!
:::

:::{.proposition title="Modules over Noetherian rings are finitely-generated"}
$R$ is Noetherian (resp. Artinian) iff every $M\in \rmod^\fg$ is Noetherian (resp. Artinian).
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:::{.remark}
We'll use this later to find a finitely generated $\ZZ\dash$module containing $\OO_K$ for $K$ a number field, so $\OO_K \in \zmod^\fg$ as a submodule of a module over a Noetherian ring.
Note that this is much easier than actually finding a $\ZZ\dash$basis of $\OO_K$.
:::

:::{.theorem title="Akuzuki-Hopkins"}
$R$ is Artinian iff $R$ is Noetherian and dimension zero.\autocite[Theorem 8.35]{CA}
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:::{.remark}
In Artinian rings, all primes are maximal.
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:::{.example title="?"}
In $\ZZ$, one can take $\gens 2 \contain \gens 4 \contains \gens 8 \contains \cdots$.
In any domain $R$ which is not a field, if $x\in R$ is a nonzero nonunit, one can take $\gens x \contains \gens {x^2} \contains \gens{x^3} \contains \cdots$.
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:::{.example title="?"}
Any residue ring of a Dedekind domain will be Artinian, e.g. $\ZZ/N\ZZ$.
In fact such rings will have finitely many ideals.
Of course any finite ring is automatically Artinian.
:::

:::{.exercise title="?"}
Show that $\CC[x,y]/\gens{x, y}^2$ is Artinian and has continuum-many ideals.
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:::{.remark}
Note that $\ZZ/N\ZZ \cong \prod \ZZ/p_i^{a_i} \ZZ$, where the latter rings are simpler since the lattice is totally ordered: ideals in $\ZZ/p^n\ZZ$ correspond to ideals in $\ZZ$ containing $p\ZZ$, so we get a chain $\gens 0 =\gens{p^n} \subset \gens{p^{n-1}} \subset \cdots\subset \gens{p} \subset \gens{1}$.
Note the unique maximal ideal $\gens{p}$, making this a **local ring**, recalling that $R$ is local iff $\size \mspec R = 1$.
If $N = \prod_{i=1}^r p_i^{a_i}$ with $r\geq 2$, then $\mspec \ZZ/N\ZZ = \ts{ \gens{p_i}}_{i=1}^r$, so $\ZZ/N\ZZ$ is only local when $r=1$.
:::

:::{.exercise title="?"}
Show that any ideal of a product is a product of ideals, i.e. if $I\normal \prod R_i$ then $I = \prod_{i=1}^r I_i$ with $I_i \normal R_i$.
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:::{.theorem title="A nice theorem"}
Every Artinian ring $R$ is a finite product of local Artinian rings $R = \prod_{i=1}^r R_i$.
Thus $\size \mspec R = r$.
:::


## Annihilators

:::{.remark}
Recall:

- For $M\in \rmod$, annihilators are defined as $\Ann_R(m) = \ts{r\in R \st rm = 0_M}\normal R$.
- $M$ is torsionfree iff $\Ann_R(m) = 0$ for all $m\neq 0$ -- this is only defined over $R$ a domain.
- Structure theorem: for $R$ a PID, $M\cong \bigoplus _{1\leq i\leq n} R/\gens{x_i} \cong M_\tors \oplus R^N$ for some $N$.
  - If $M$ is finitely-generated and torsionfree over a PID, $M$ is free.
- Note that torsionfree is always a necessary condition for freeness in any case.
- Dedekind domains generalize PIDs.
- For $S\subseteq M$, define $\Ann_R(S) = \ts{r\in R \st rs = 0\,\, \forall s\in S} = \Intersect _{s\in S} \Ann_R(s) \normal R$.
- $M$ is faithful iff $\Ann_R(M) = 0$
  - Note that the $R\dash$module structure is group morphism $R\to \Endo_\ZZ(M)$ where $r\mapsto r\cdot$, so $M$ is faithful iff this is injective.
- $M$ is cyclic if $M = Rm$ for a single element $m\in M$.
- $M$ is simple iff $M\neq 0$ and has no proper nontrivial submodules.
:::

:::{.exercise title="?"}
Show that any $M\in\rmod$ canonically determines a faithful module $M \in \mods{R/\Ann_R(M)}$.
:::

:::{.exercise title="?"}
Show that 

- If $M$ is a cyclic module then $M\cong R/\Ann_R(M)$.
- If $M$ is simple then there exists a unique $\mfm\in \mspec R$ with $M\cong R/\mfm$.

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## Composition series

:::{.remark}
More review:

- For $G\in\Fin\Grp$, a Jordan-Hölder series is a chain of subgroups, each normal in the next, where each successive quotient is simple.
  If the quotients were not simple, one could include additional factors, so this is equivalent to being a maximal chain.
- Big theorem: the isomorphism types and multiplicity (so the multiset) of the Jordan-Hölder factors are uniquely determined, independent of the choice of series.
  - Example: for $G = (\ZZ/N\ZZ, +)$, the factors are $\ZZ/p_i \ZZ$ with multiplicity $a_i$.
- For modules, there is a close analogue: a JH series for $M\in\rmod$ is a chain of submodules with simple quotients.
- Theorem: the JH theorem holds verbatim, provided a JH series exists.
  - A JH series for $M$ exists iff $M$ is both Artinian and Noetherian, e.g. if $R$ is Artinian and $M$ is finitely-generated.
- The length of $M$ is the length of any JH series.

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:::{.remark}
Next:

- Projective modules
- Localization
- Fractional ideals (and invertible fractional ideals)

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