\part{Quotient rings}

\chapter{Ideals}

Remember quotient groups? It's quite nice that cosets partition a group (thanks Lagrange); this along with the ability of normal subgroups to make our quotient map a well defined group homomorphism allowed the creation of a group on these 'divided' equivalence classes.
In group theory, quotient groups must be formed from normal subgroups to ensure that the quotient group operation functions as desired. One may ask; what's the ideal (pun intended) subring that can be used to make a notion of a 'quotient ring'?

Unfortunately, subrings won't do the the trick, but if we want to emulate the idea of left multiplication cosets, we can form some kind of set that would ideally (this joke is going to get seriously overused in this part) do the trick.

\begin{definition}
Let $(R,+,\cdot)$ be a ring. A \emph{two-sided ideal of $R$} is a subgroup $I$ of $(R,+)$ closed under addition, multiplication, and multiplication from any element in $R$. For brevity, two-sided ideals will just be called ideals.
\[r \in R,i,j \in I\]
\[i+j \in I\]
\[ ij \in I \]
\[ ri \in I\]
\[ ir \in I\]
\end{definition}

One can define specific ideals where arbitrary ring multiplication is allowed only from the left or right.

\begin{definition}
Let $(R,+,\cdot)$ be a ring. A \emph{left ideal of $R$} is a subgroup $I$ of $(R,+)$ closed under addition, multiplication, and left multiplication from any element in $R$.
\[r \in R,i,j \in I\]
\[i+j \in I\]
\[ ij \in I \]
\[ ir \in I\]
\end{definition}

We'll focus our attention on two-sided ideals from now on.



\begin{proposition}
If $1_R \in I$, then $I=R$.
\end{proposition}


\begin{proposition}
Let F be a field, only ideals are $\{0_R\}$ and $F$.
\end{proposition}


\[I \cap J\]
\[I \oplus J\]
\[ \langle ij : i\in I , j \in J \rangle \]

\[ \langle r_1 r_2 , \hdots , r_n \rangle = \{ \sum^{n}_{k=1} \lambda_k r_k  : \lambda_k \in R \} \]

\[ \langle f : f \in M \rangle = \{ \sum_{f_k \in N} \lambda_k f_k  : \lambda_k \in R, N \subseteq M, |N|< \infty \} \]
Ideals equal to some are 'finitely generated'.

\section{Basic types of ideals}

\begin{definition}
$\{0_R\}$ is the \emph{zero ideal}.
\end{definition}

Every ring has the zero ideal; it's trivial and lame so we won't have much to say about it. We've proven that when the identity is in our ideal, the ideal must be the whole ring. This is another trivial ring that we may as well mention for completeness.

\begin{definition}
Let $(R,+,\cdot)$ be a ring, then $R$ is the \emph{unit ideal}.
\end{definition}


\begin{definition}
A \emph{principal ideal} is an ideal of the form $\langle d \rangle$. In other words, it is generated by one element.
\end{definition}

If D is a domain where all its ideals are principal ideals, PID (principal ideal domain)

\begin{example}
$(\mathbb{Z},+,\cdot)$ is a PID and all its ideals are $d\mathbb{Z}$. This is because the only subgroups of $(\mathbb{Z},+)$ are of the form $d\mathbb{Z}$, and these can be verified to be principal ideals since addition is obviously satisfied (being a subgroup), multiplication in and out of the ideal is satisfied since $r(dn)=d(rn)\in d\mathbb{Z}$.
Note that $0\mathbb{Z}$ corresponds to the zero ideal.
\begin{example}


\chapter{Quotient rings}

\section{Quotient rings}
\begin{definition}
A \emph{quotient ring on $R$} is a ring of unique left 'addition' cosets on the ideal $I$. Addition and multiplication are defined in the following way.
\[ ( R / I , \oplus, \otimes )\]
\[ (r+I) \oplus (s+I) = (r+s)+I\]
\[ (r+I) \otimes (s+I) = rs+I\]
\end{definition}

We should make sure that $\oplus, \otimes$ are well defined operations; we didn't introduce ideals for nothing!


For the sake of simplicity, we'll often use the equivalence class notation $[r]=r+I$ with the original ring's  addition and multiplication symbols $+,\cdot$ for elements, addition, and multiplication of a quotient ring. Here's an example below.

\[ (r+I) \otimes ((s+I) \oplus (t+I)) = [r]([s]+[t]) \]

\section{Examples of quotient rings}
\section{Prime ideal}

One may want to know when a quotient ring is a domain; 

Recall that domains are categorized by the fact that $xy=0_D$ iff either $x,y$ equals $0_D$. $[0_D]$ is the left coset representing the domain itself, so assuming our quotient ring is a domain, when $[x][y]=[0_R]$, either $[x]$ or $[y]$ equals $[0_R]$; meaning that one of them is in the same coset (the ideal) as $0_R$.

This allows us to forge precisely the right ideal that makes a quotient ring a domain.

\begin{definition}
A \emph{prime ideal $I$} is an ideal such that for any ring elements $r,s$, if $rs$ is in the ideal, then either $r$ or $s$ is in the ideal.
\end{definition}

Note that this looks slightly similar to Euclid's lemma!


\begin{proposition}
$R / I$ is a domain iff $I$ is a prime ideal.
\end{proposition}

\section{Maximal ideal}

Similarly, one wants to know which type of ideal forms a field.

\begin{definition}
A \emph{maximal ideal $I$} is an ideal such that any ideal $J$ where $I \subset J$ equals the entire ring. It is the largest non-unit ideal possible on a ring.
\end{definition}


\begin{proposition}
$R / I$ is a field iff $I$ is a maximal ideal.
\end{proposition}
