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\hfill \textbf{February 7, 2012}

\vspace{0.5 cm} \centerline{ \textbf{\large{Homework 2\hspace{1 cm}
Math 6303}}}

\vspace {1 cm}



Let $\mbf{M}$ be a module over the ring $\mbf{A}$ and let $\mbf{M}_i$ be $\mbf{A}-$submodules, $i=1,\ldots, n$ . Then $\mbf{M}$ is
called the \textit{direct sum} of the $\mbf{M}_i$ if every $m\in \mbf{M}$ is a unique sum of elements $m_i\in \mbf{M}_i$. Notation
\[
\mbf{M}=\mbf{M}_1\oplus\cdots\oplus \mbf{M}_n
\]
Prove:

\begin{enumerate}
\item
 $\mbf{M}=\mbf{M}_1\oplus\cdots\oplus \mbf{M}_n$ iff $\mbf{M}_i
\cap (\mbf{M}_1\oplus \cdots \oplus \mbf{M}_{i-1}\oplus
\mbf{M}_{i+1}\oplus \cdots \oplus \mbf{M}_n)=\mbf{O} $ iff Given
any $\mbf{A}$--module $\mbf{N}$ and $\mbf{A}$--module
homomorphisms $\varphi_i:\mbf{M}_i \to \mbf{N}$ then there is a
unique $\mbf{A}$--module homomorphism $\varphi: \mbf{M} \to
\mbf{N}$ which extends the $\varphi_i$.
\item
$\mbf{M}/\mbf{M}_i\cong \mbf{M}_1\oplus \cdots \oplus \mbf{M}_{i-1}\oplus
\mbf{M}_{i+1}\oplus \cdots \oplus \mbf{M}_n$


\item
Let $\mbf{P}$ be a projective $\mbf{A}-$module and
$\mbf{P}=\mbf{M}\oplus \mbf{N}$. Then $\mbf{M}$ and $\mbf{N}$
are also projective.  Conclude that $\mbb{Z}_3$   is a
projective module over the ring $\mbb{Z}_6$ and that there are
projective modules that are not free.
\item
Prove that a projective $\mbb{A}-$module is a direct summand of a
free  $\mbb{A}-$module.
\item  Prove that a finitely generated
projective module over a principal ideal domain $\mbf{D}$ is free.




\end{enumerate}







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