\documentclass[12pt]{amsart}
\title{Algebra 2. Homework 6}
\author{}

\usepackage{amsmath,amssymb}

\usepackage[margin=1in]{geometry}
\usepackage[all]{xy}
\xyoption{arc}
\newcommand{\ds}{\displaystyle}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}

\newcommand{\CoComp}[1]{\mathbb{K}^\bullet\left(#1\right)}
\newcommand{\Comp}[1]{\mathbb{K}_\bullet\left(#1\right)}

\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\ul}[1]{\underline{#1}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\wt}[1]{\widetilde{#1}}
\newcommand{\wh}[1]{\widehat{#1}}
\newcommand{\morph}[1]{\stackrel{#1}{\longrightarrow}}

\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}

\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\Coker}{\operatorname{CoKer}}
\newcommand{\Coim}{\operatorname{CoIm}}

\newcommand{\CalA}{\mathcal{A}}
\newcommand{\CalB}{\mathcal{B}}
\newcommand{\CalC}{\mathcal{C}}
\newcommand{\CalD}{\mathcal{D}}

\newcommand{\ga}{\mathfrak{a}}

\newcommand{\Ob}{\operatorname{Ob}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ext}{\operatorname{Ext}}
\newcommand{\Tor}{\operatorname{Tor}}

\newcommand{\End}{\operatorname{End}}
\newcommand{\Bil}{\operatorname{Bilinear}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\triv}{\mathbf{0}}
\newcommand{\h}{\mathsf{h}}
\newcommand{\op}{\scriptscriptstyle{\operatorname{op}}}
\newcommand{\fd}{\scriptscriptstyle{\operatorname{fd}}}


\newcommand{\Cov}[2]{{\mathcal{F}}(#1,#2)}
\newcommand{\Contra}[2]{{\mathcal{F}}(#1^{\op},#2)}

\newcommand{\Sets}{{\mathbf{Sets}}}
\newcommand{\Gps}{{\mathbf{Gps}}}
\newcommand{\Ab}{{\mathbf{Ab}}}
\newcommand{\Vect}{{\mathbf{Vect}}}
\newcommand{\GSets}[1]{#1\mbox{-}\Sets}
\newcommand{\Rmod}[1]{#1\mbox{-}\mathbf{\operatorname{mod}}}
\newcommand{\modR}[1]{\mathbf{\operatorname{mod}}\mbox{-}#1}

\newcommand{\isom}{\stackrel{\sim}{\longrightarrow}}

\newcommand{\Dlim}[1]{\ds \lim_{\begin{subarray}{c}
\longrightarrow \\ #1\end{subarray}}}
\newcommand{\Ilim}[1]{\ds \lim_{\begin{subarray}{c}
\longleftarrow \\ #1\end{subarray}}}

\newcommand{\Ccy}{\mathcal{Z}}
\newcommand{\Cb}{\mathcal{B}}

\newcommand{\problem}[1]{\noindent {\bf Problem #1.--}}

\begin{document}
\maketitle

\begin{center}
$\Rmod{R}$ is the abelian category
of left $R$--modules, over a unital ring $R$.\\
\end{center}

\hrule

\vspace*{0.2in}

\problem{1} Let $I$ be a set and let $R^{(I)}$ be the direct sum
of $I$ copies of $R$, viewed as a left $R$--module.
\[
R^{(I)} = \bigoplus_{i\in I} M_i,\qquad \text{where }
M_i = R,\ \forall i\in I.
\]
Prove that $R^{(I)}$ is a projective module. {\em Any module
isomorphic to $R^{(I)}$ for some indexing set $I$ is called
a free $R$--module. This problem is asking you to prove that
free implies projective.}\\

\problem{2} Prove that an $R$--module $P$ is projective if,
and only if there exists $P'$ such that $P\oplus P'$ is free
(see problem 1).\\

\problem{3} Prove that $\Rmod{R}$ has enough projectives,
using Problem 1.\\

\problem{4} Assume that there exists $p\in R$ such that $p^2=p$.
Prove that $M=Rp$ (left ideal generated by $p$) is a projective
$R$--module.\\

\problem{5} Prove that $\Z/3\Z$, is a projective $\Z/6\Z$--module.\\

\problem{6} Let $M\in\Rmod{R}$ and assume that we have two
short exact sequences with $P_1,P_2$ projectives:
\[
0\to N_1 \to P_1\to M\to 0, \qquad\text{and}\qquad
0\to N_2\to P_2\to M\to 0.
\]
Prove that $P_1\oplus N_2 \cong P_2\oplus N_1$.\\

\problem{7} Prove that direct sum of projective modules is projective.
Prove that direct product of injective modules is injective.\\

\problem{8} Assume that $R$ is commutative, and $P$ is a projective
$R$--module. Prove that $P\otimes-:\Rmod{R}\to\Rmod{R}$ is exact.\\

\problem{9} Let $N\in\Z_{\geq 2}$. Verify that $\ds 0\to \Z \morph{\mu_N} \Z \to 0$,
where $\mu_N$ is multiplication by $N$, is a projective resolution
of $\Z/N\Z\in\Rmod{\Z}$.\\

\problem{10} Prove that the following infinite complex is
a projective resolution of $\Z/3\Z$ as a module over $\Z/9\Z$:
\[
\cdots\morph{\mu_3} \Z/9\Z \morph{\mu_3} \Z/9\Z \morph{\mu_3} \Z/9\Z \to 0.
\]
\ \\

\problem{11} Assume that $R$ is commutative, and $M,N\in\Rmod{R}$.
Let $Q\in\Ab = \Rmod{\Z}$. Prove that we have an isomorphism
of $R$--modules:
\[
\Hom_R(M,\Hom_{\Z}(N,Q)) \cong \Hom_{\Z}(M\otimes_R N, Q),
\]
where, for an $R$--module $X$ and an abelian group $Y$, the abelian
group $\Hom_{\Z}(X,Y)$ has a structure of an $R$--module via:
\[
r\in R,\ f\in\Hom_{\Z}(X,Y)\qquad\rightsquigarrow\qquad
(r\cdot f)(x) = f(rx).
\]

\end{document}
