# October 27th

## Modules

Let $R$ be a ring and $M$ be an $R\dash$module.

**Definition:**
For a subset $X\subseteq M$, we can define the *submodule generated by $X$* as
$$
\generators{X} \definedas \intersect_{X \subseteq N \leq M} N \subseteq M
.$$

Then $M$ is generated by $X$ iff $M = \generators{X}$.

As a special case, when $X = \theset{m}$ consists of a single element, we write
$$
\generators{m} = Rm \definedas \theset{rm \suchthat r\in R}
.$$

In general, we have
\begin{align*}
\generators{X} =  \theset{\sum r_i x_i \mid r_i \in R, x_i \in X}
.\end{align*}

## Direct Products and Direct Sums


**Definition:**
Let $\theset{M_i}$ be a finite collection of $R\dash$modules, and let
\begin{align*}
N = \bigoplus M_i = \theset{\sum m_i \mid m_i \in M_i}
\end{align*}

with multiplication given by $\gamma \sum m_i = \sum \gamma m_i$ denote the **direct sum**.

For an infinite collection, we require that all but finitely many terms are zero.

**Definition:**
Define $N = \prod M_i$ denote the **direct product**, where we now drop the condition that finitely many terms are zero.

When the indexing set is finite, $\bigoplus M_i \cong \prod M_i$.
In general, $\bigoplus M_i \injects \prod M_i$.

Note that the natural inclusions 
$$
\iota_j: M_j \injects \prod M_i
$$

and projections
$$
\pi_j: \prod M_i \surjects M_j
$$

are both $R\dash$module homomorphisms.

Theorem:
$M \cong \bigoplus M_i$ iff there exist maps $\pi_j: M \to M_j$ and $\iota_j: M_j \to M$ such that

1. 
$$
\pi_j \circ \iota_k = \begin{cases} 1m & j=k \\ 0 & \text{else}\end{cases}
$$

2. $\sum_j \iota_j \circ \pi_j = \id_M$

**Remark:**
Let $M, N$ be $R\dash$modules. 
Then $\hom_{R-\text{mod}}(M, N)$ is an abelian group.


## Internal Direct Sums

For a collection of submodules of $M$ given by $\theset{M_i}$, denote the *internal direct sum*
\begin{align*}
\sum M_i \definedas \theset{m_1 + m_2 + \cdots \mid m_i \in M_i}
\end{align*}

iff it satisfies the following conditions:

1. $M = \sum_i M_i$

2. $M_i \intersect M_j = \theset{0}$ for $i\neq j$.


## Exact Sequences

**Definition:**
A sequence of the form
$$
0 \to M_1 \mapsvia{i} M_2 \mapsvia{p} M_3 \to 0
$$

where

- $i$ is a monomorphism

- $p$ is an empimorphism

- $\im i = \ker p$

is said to be **short exact**.

*Examples*:

- 
$$
0 \to 2\ZZ \injects \ZZ \surjects \ZZ/2\ZZ \to 0
$$

- For any epimorphism $\pi: M\to N$,
$$
0 \to \ker \pi \to M \to N \to 0
$$

- 
$$
0 \to M_1 \to M_1 \oplus M_2 \to M_2 \to 0
$$


In general, any sequence
$$
\cdots \to M_i \mapsvia{f_i} M_{i+1} \mapsvia{f_{i+1}} \cdots
$$

is **exact** iff $\im f_i = \ker f_{i+1}$.
 

1. If $\alpha, \gamma$ are monomorphisms then $\beta$ is a monomorphism.