# Friday April 3rd

Recall that the definition of a normal series for $G$ a group.

Theorem (Jordan Holder)
: Any two composition series for the same group $G$ are equivalent (same isomorphism classes of quotients and multiplicities).

There is an analog of this for modules, even over a noncommutative ring:  this is just a sequence of submodules inclusions, since normality is automatic.
There is similarly a notion of Schreir refinement.
For $p$ groups, the composition factors have to be cyclic of order $p$.
On one hand, we could fix the series and ask for what modules have a compatible composition series -- this is the extension problem, and is difficult in general.
Here we will fix the module and see what the possible composition series are.

Question:
When does an $R\dash$ module admit a finite composition series?

Answer:
When $R$ is both Noetherian and Artinian.


Suppose that $M$ satisfies the ACC and DCC.
Then there exists a minimal simple module $M_1 < M$, an $M_2$ properly containing $M_1$ such that $M_2/M_1$ is simple, and so on.
This sequence of inclusions terminates due to the ACC, so this yields a finite composition series.

Definition (Length)
: ?

Proposition (Length is Additive over SESs)
: For $0 \to M' \to M \to M'' \to 0$, $M$ has finite length iff $M', M''$ do, and $\ell(M) = \ell(M') + \ell(M'')$.

> Dream of commutative algebra: every theorem at the level of generality of "Let $M$ be a module over a Noetherian ring".

Proposition (Quotient/Localization )f Noetherian is Noetherian)
: Quotients and localizations of Noetherian rings are Noetherian.

Proposition (Ideal Poset of Quotient Order-Injects into Ideal Poset of Full Ring)
: For $I\normal R$, $\mci(R/I) \injects \mci(R)$ is an isotone inclusion of posets.

Thus Noetherian-Artinian properties in the RHS imply the same properties in the LHS.
For localizations, we also have $\mci(S\inv R) \injects \mci(R)$ by push-pull properties.

Proposition (Artinian Domains are Fields)
: If $R$ is an Artinian domain, then $R$ is a field.

Proof
:   For the contrapositive, let $a\in R^\bullet \setminus R\units$, then $(a) \supsetneq (a^2) \supsetneq \cdots$ is an infinite descending chain.

Theorem (Properties of Artinian Rings)
:   For $R$ Artinian,

    a. $\dim_{\text{Krull}} R = 0$.
    b. $\mcj(R) = \nil(R)$.
    c. $\maxspec R = \theset{\mfm_i}^n_{i=1}$ is finite.
    d. $\nil(R)$ is a nilpotent ideal.

Proof
:   \hfill
    a. If $\mfp \in \spec k$, $R/\mfp$ is an Artinian domain and thus a field, so $\mfp$ is maximal.
    b. Produce a descending chain $\mfm_1 \supset \mfm_1 \mfm_2 \supset \cdots$ and suppose that $\prod^n \mfm_i = \prod^{n+1} \mfi$, then $\prod^n \mfm_i \subset \mfm_{n+1}$ and thus $\mfm_{n+1} \supset \mfm_i$ for some $i$, which is a contradiction.
    c. ?
    d. ?

Theorem (Akizuki-Hopkins)
: $R$ is Artinian $\iff R$ is Noetherian and $\dim R = 0$.