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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#18}
\date{Assigned Friday, 20 October 2017; due Monday, 23 October 2017}

\maketitle

\vspace{1ex}

\begin{problab}{18.1} 
Let $R$ be a (commutative integral) domain.  An $R$-module $A$ is \emph{divisible} if $rA=A$ for every nonzero $r \in R$.

Let $Q$ be a nonzero divisible $\Z$-module.  Prove that $Q$ is \emph{not} a projective $\Z$-module.  Deduce that $\Q$ is not a projective $\Z$-module.  \emph{[Hint: Show first that if $F$ is a free module then $\bigcap_{n=1}^{\infty} nF = \{0\}$ using a basis.  Suppose that $Q$ is projective, and use one of the equivalent conditions on a projective module.]}
\end{problab}

\begin{problab}{18.2}
Let $R$ be a commutative ring.  Let $M,N$ be projective $R$-modules.  Show that $M \otimes_R N$ is a projective $R$-module.  \emph{[Hint: Use that the tensor product of two free $R$-modules is free, because tensor products commute with direct sums.]}
\end{problab}

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