\part{Ring homomorphisms}

\chapter{Ring homomorphism}
\section{Properties of ring homomorphism}
\section{Examples of ring homomorphism}

\section{Ring monomorphism}
\section{Ring epimorphism}
\section{Ring endomorphism}

These types of ring homorphisms have particularly interesting properties, but ring isomorphisms are perhaps the most important class of homomorphisms.

\chapter{Ring isomorphism}
\section{Properties of Ring isomorphism}
\section{Examples of Ring isomorphism}
\section{Ring automorphism}



Some Theorems

A \emph{Canonical map} is a function between two objects that arises from their definitions. It is a function used to define the behaviour of some object.

Kernel (Ring theory)

\[ \mathrm{ker}(f)= \{r \in R : f(r)=0_S \}\]
Image

Isometry


\section{First isomorphism theorem}


We can create quotient groups given any normal subgroup. Remember that kernels and images of homomorphisms actually form subgroups themselves; that's pretty neat. However can one make quotient groups with the kernel?


\begin{lemma}
\[ f : G \to H \text{ is a homomorphism } \implies \mathrm{ker}(f) \lhd G \]
\end{lemma}

This means we can make quotient groups using kernels. So the cosets on the kernel form their own group. Experimenting with the cosets of the kernel brings the following lemma.

\begin{lemma}
\[ f : G \to H \text{ is a homomorphism } \implies [ g_1 \mathrm{ker}(f) = g_2 \mathrm{ker}(f) \iff f(g_1)=f(g_2) ] \]
\end{lemma}

One way of interpreting this lemma is that kernel cosets correspond to sets of elements with the same homomorphism mappings. This is the essence of the \emph{first isomorphism theorem}.

\begin{theorem}[First isomorphism theorem for groups]
Let $f : G \to H$ be a homomorphism . Then $\mathrm{Im}(f) \cong G / \mathrm{ker}(f)$ the following isomorphism $k$.
\[ k : g \mathrm{ker}(f) \mapsto f(g)\]
\begin{tikzcd}[column sep=small]
	G \arrow{r}{f} \arrow{d}{\pi} & \mathrm{Im}(f) \\
G/\mathrm{ker}(f) \arrow{ru}{k}
\end{tikzcd}
\end{theorem}

\begin{proposition}
For any ring $R$, the following $ f : \mathbb{Z} \to R$ is a ring homomorphism.
\[f(n) = \sum^{n}_{i=1} 1_R \]
\end{proposition}

\chapter{Ring characteristic}

\begin{definition}
\[\mathrm{char}(R) = \mathrm{ord}_{+}(1_R)\]
\end{definition}

Let $D$ be a domain, then $\mathrm{char}(D)$ is either prime or 0.
