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\title{MAT 250B HW07}
\author{[add your name here]}
\date{Due Friday, 2/23/24 at 11:59 pm on Gradescope}

\begin{document}

\maketitle




%%%%%%%%%%%%%%%%%%%%%%
\exercise
Recall that an abelian group $A$ is \textbf{torsion-free} if it has no nonzero elements of finite order.
%
Prove that the following conditions are equivalent for an abelian group $A$:
\begin{enumerate}[label={(\roman*)}]
    \item $A$ is torsion-free and divisible.
    \item For every $n \in \N$\footnote{where $\N = \Z_{>0}$}, the multiplication map $\mu_n: A \to A$, $a \mapsto na$, is an isomorphism.
    \item $A$ is a vector space over $\Q$.
    
\end{enumerate}
 


%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $A$ be a torsion-free abelian group.
Prove that $A$ can be embedded as a subgroup of a vector space over $\Q$. \textit{Hint: Embed $A$ in a divisible abelian group $D$, and then show that $A \cap T(D) = \{0\}$, where $T(D)$ is the torsion subgroup of $D$.}


%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $\{A_i\}_{i \in I}$ be an arbitrary collection of right $R$-modules. Prove that $\bigoplus_{i \in I} A_i$ is flat if and only if every $A_i$ is flat.


%%%%%%%%%%%%%%%%%%%%%%
\exercise

In this exercise you may use the following proposition:
\begin{proposition}
    Let $k$ be a PID. Then any submodule of a free $k$-module is also free.
\end{proposition}

\bea

\item Prove that over a PID, a module $M$ is projective if and only if it is free.

\item Prove that as a $\Z$-module, $\Q$ is flat but not projective.


\ee












\end{document}
