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  {\large Mathematics 111}\\Spring 2007\\Homework 1
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Prove the following theorem (Proposition 2.2, Chapter 3 of Lang) which
is analagous to the ``contravariant'' one proved in class.

\bigskip

\noindent\textbf{Theorem:}  Let $A$ be a ring with identity.  The sequence of
left $A$-modules
\[\xymatrix{%
  0 \ar@{>}[r]& N' \ar@{>}[r]^\varphi &N \ar@{>}[r]^\psi&N''}
\]
is exact if and only if for all left $A$-modules $M$, the sequence
\[\xymatrix{0 \ar@{>}[r]&\Hom_A(M,N') \ar@{>}[r]^{\varphi_*}&
  \Hom_A(M,N) \ar@{>}[r]^{\psi_*}& \Hom_A(M,N'')}\]
is exact.





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\noindent\textit{Hint:} For the converse, a single choice of $M$ can
work for all parts, but you still need to sweat the details.
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