\section{Modules}
\subsection{Tensor product}
\begin{Definition}
    Let $R$ be a ring and $M, N$ be $R$-modules.
    A \emph{tensor product of $M$ and $N$} is a pair $(T, \phi)$, where
    \begin{itemize}
        \item $T$ is a $R$-module and
        \item $\phi: M \oplus N \to T$ is $R$-bilinear,
    \end{itemize}
    and the following universal property is fulfilled:

    If $L$ is a $R$-module and $\chi: M \oplus N \to T$ is $R$-bilinear, there is a unique $R$-homomorphism 
    $g: T \to L$ such that $\chi = g \circ \phi$.
\end{Definition}

\begin{Lemma}
    Let $R$ be a ring and $M, N$ be $R$-modules.
    \begin{enumerate}
        \item The tensor product of $M$ and $N$ is unique up to isomorphism.
        \item The tensor product of $M$ and $N$ exists.
    \end{enumerate}
\end{Lemma}