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\def\title{Homework 2, January 24}
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\bs
These problems are due Friday, February 2.  Read Chapter X, section 1.  Do
problems 4, 5, 8a, and 9, on page 476.  Also do the following problems.

\bs
{\bf Problem 1.}  Suppose $f : R \to S$ is a ring homomorphism of commutative rings.  Then we may regard $S$ as a
$R$-module by defining $r \cdot s$ to be $f(r)s$.  Suppose also that $g : R
\to T$ is a ring homomorphism.  Then the tensor product $S \otimes_R T$ becomes
a commutative ring with unit when we define the product $(s \otimes t) \cdot (s' \otimes
t')$ to be $(s s')\otimes(t t')$; check that.  Produce ring homomorphisms $S \to S
\otimes_R T$ and $T \to S \otimes _R T$ which make the square
$$
\CD
	R	@>f>>	S	\\
	@VgVV		@VVV	\\
	T	@>>>	S \otimes_R T
\endCD
$$
commute.  Show that this square is a pushout in the sense of category theory,
which means that, given any commutative square
$$
\CD
	R	@>f>>	S	\\
	@VgVV		@VVV	\\
	T	@>>>	U
\endCD
$$
in the category of commutative rings, there is a unique ring homomorphism
$S \otimes_R T \to U$ which makes the triangles
$$
\CD
	S	@>>>	S \otimes_R T \\
	   &  \searrow & \downarrow   \\
	@.		U
\endCD
$$
and
$$
\CD
	T	@>>>	S \otimes_R T \\
	   &  \searrow & \downarrow   \\
	@.		U
\endCD
$$
commute.

\bs
{\bf Problem 2.}  The ring $\C \otimes_{\R} \C$ is four-dimensional as an
$\R$-vector space.  Identify this ring.



\bye


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