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\begin{document}

\noindent
{\large \bf Additional notes \#4.}

\bigskip

Read Section 10.4.

\bigskip

\noindent
{\bf Theorem.} If $d = \gcd(m,n)$,
then ${\BZ}_m\otimes_{\BZ} {\BZ}_n\cong {\BZ}_{d}$.

\medskip

{\it Proof.} Define a homomorphism $\varphi: {\BZ}\rightarrow 
{\BZ}_m\otimes_{\BZ} {\BZ}_n: 1\mapsto 1\otimes 1$.
Since ${\BZ}_m\otimes_{\BZ} {\BZ}_n$ is generated by
the elements of the form $m\otimes n = 1\otimes mn = 
mn(1\otimes 1) = \varphi(mn)$, it follows that $\varphi$
is onto. Since $\varphi(m) = m\otimes 1 = 0\otimes 1 = 0\otimes
0$, and $\varphi(n) = 1\otimes n = 1\otimes 0 = 0\otimes 0$,
we get that $\ker(\varphi)$ contains $(m,n) = (d)$.
By the first isomorphism theorem there is an
onto homomorphism $\overline{\varphi}:{\BZ}_d\rightarrow
{\BZ}_m\otimes_{\BZ} {\BZ}_n$ satisfying $\overline{\varphi}(1)
= 1\otimes 1$.

In the other direction, the function 
$\psi: {\BZ}_m\times {\BZ}_n\rightarrow {\BZ}_d: (a,b)\mapsto
ab$ is well-defined (since $d$ divides both $m$ and $n$)
and is ${\BZ}$-balanced since 
\begin{itemize}
\item $\psi(a,b+c) = a(b+c) = ab+ac = \psi(a,b) + \psi(a,c)$ in ${\BZ}_d$,
\item $\psi(a+b,c) = (a+b)c = ac+bc = \psi(a,c) + \psi(b,c)$ in ${\BZ}_d$, and
\item $\psi(ab,c) = (ab)c = a(bc) = \psi(a,bc)$ in ${\BZ}_d$.
\end{itemize}
Thus $\psi$ extends uniquely to a homomorphism 
$\overline{\psi}: {\BZ}_m\otimes_{\BZ} {\BZ}_n
\rightarrow {\BZ}_d$. Since $\overline{\psi}(1\otimes 1) = 1$,
we get that $\overline{\psi}\circ \overline{\varphi}$
is a homomorphism of ${\BZ}_d$ to itself which maps 
the generator $1$ to itself. 
The identity function is the only homomorphism 
that can do this (see Exercise~(3) below), hence
$\overline{\psi}\circ \overline{\varphi} = {\rm id}$.
A similar argument shows that 
$\overline{\varphi}\circ \overline{\psi} = {\rm id}$.
Hence 
${\BZ}_m\otimes_{\BZ} {\BZ}_n\cong {\BZ}_{d}$. \hfill $\Box$

\bigskip\bigskip

\noindent
{\bf Practice Exercises on Tensor Products.} 
\begin{enumerate}
\item[(1)] Let $\m r$ be a commutative ring.
Show that if $I$ and $J$ are ideals of $\m r$, then
the $\m r$-module 
${\m r}/I\otimes_{\m r} \m r/J$ is isomorphic to
$\m r/(I+J)$.
\bigskip
\item[(2)] Let $\BF$ be a field, and let $M_n({\BF})$
be the ${\BF}$-algebra of all $n\times n$-matrices
over $\BF$. Show that the ${\BF}$-algebra
$M_m({\BF})\otimes_{\BF} M_n({\BF})$ is isomorphic
to $M_{mn}({\BF})$. (Hint: if you can figure out the case
$m = n = 2$, then you can probably figure out the general
case.)
\end{enumerate}

\bigskip

\noindent
{\bf General Practice Exercises.} 
\begin{enumerate}
\item[(3)] Suppose that $G$ is a generating set for
the algebra $\m a$. Show that the only homomorphism
$\varphi:\m a\rightarrow \m a$ that is the identity
on $G$ is the identity function. (Hint: do the
next problem first.)
\item[(4)] Suppose that $\varphi, \psi:\m a\rightarrow \m b$
are homomorphisms. Show that the {\it equalizer}
of $\varphi$ and $\psi$, defined by $\lv\varphi=\psi\rv =
\{a\in A\;|\;\varphi(a) = \psi(a)\}$, 
is a subalgebra of $\m a$. (Now do the previous
problem with $\m b = \m a$ and $\psi = {\rm id}_{\m a}$.)
\end{enumerate}

\end{document}

