\documentclass[11pt]{article}
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\usepackage[margin=1in]{geometry}
\usepackage{enumitem}
\usepackage{amsmath, amssymb, amsthm, mathrsfs}
\usepackage{tikz, tikz-cd}
\usetikzlibrary{decorations.markings}
\usepackage{xcolor}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{hyperref}
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    colorlinks=true,
    linkcolor=blue,
    filecolor=magenta,      
    urlcolor=blue,
    pdftitle={Algebra},
    pdfpagemode=FullScreen,
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% THEOREM ENVIRONMENTS
\theoremstyle{definition}
\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}



% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real numbers
\newcommand{\C}{\mathbb{C}} % complex numbers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
% \newcommand{\st}{ \ : \ } % such that
\newcommand{\st}{ \ | \ } % such that
\newcommand{\inv}{^{-1}} % inverse
\newcommand{\into}{\hookrightarrow}  % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
\newcommand{\id}{\mathrm{id}} % identity map
\newcommand{\Frac}{\mathrm{Frac}} % fraction field
\newcommand{\normal}{\unlhd} % normal subgroup of
\newcommand{\F}{\mathbb{F}} % random field
\newcommand{\Isom}{\mathrm{Isom}} % isometries group

\DeclareMathOperator{\coker}{coker} % cokernel
\DeclareMathOperator{\img}{im} % image
\DeclareMathOperator{\sgn}{sgn} % sign of permutation
\DeclareMathOperator{\ord}{ord} % order of group element
\DeclareMathOperator{\lcm}{lcm} % least common multiple
% \DeclareMathOperator{\gcd}{gcd} % greatest common divisor
\newcommand{\op}{^{\mathrm{op}}} % opposite thing

\DeclareMathOperator{\Mod}{\mathbf{Mod}} % module category
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\newcommand{\ModR}{\Mod_R} % right R mods
\DeclareMathOperator{\Grp}{\mathbf{Grp}} % cat of groups
\newcommand{\SC}{\mathscr{C}} % category C
\newcommand{\CC}{\mathcal{C}} % category C (Rotman) or chain complex
\DeclareMathOperator{\Hom}{\mathrm{Hom}} % module category
\DeclareMathOperator{\Ab}{\mathbf{Ab}} % cat of abelian groups






% SOME ENVIRONMENTS
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% round matrix
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% square matrix
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\title{MAT 250B HW03}
\author{[add your name here]}
\date{Due Friday, 1/26/24 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

{
\scriptsize


\noindent \textbf{Reminder} 
Your homework submission \textbf{must be typed} (TeX'ed) up in full sentences, with proper mathematical formatting. \\

\noindent \textbf{Grading} Most problems will be graded for completion out of 2 points: 1 if you wrote something meaningful, 2 if your solution is basically correct.
A few select problems will be graded more carefully. \\

\noindent \textbf{Solutions} I will not be posting full solutions, but might post some abridged solutions for selected problems. You are responsible for looking at the solutions and determining whether your solutions are correct! This is a very important skill to have when you transition to doing research; you \textit{must} learn how to check your own work. So, if you have any questions about whether your solution is valid, come ask me or the TA (after checking my abridged solutions). \\

\noindent \textbf{Regarding figures} You are not required to \texttt{tikz} your diagrams, but you're welcome to. Feel free to hand-draw your diagrams and insert them using a \texttt{figure} environment and/or \texttt{includegraphics}.
}

\paragraph{Note} If you are working on this HW before we've covered injective modules, you should start with Exercises 2, 5a, 6, and 7.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%-----------
\exercise
Let $M \in \RMod$. 
Prove that the contravariant functor $\Hom_R(-, M)$ is left-exact. 


%-----------
\exercise
Let $P_1, P_2 \in \RMod$. Prove that $P_1 \oplus P_2$ is projective if and only if both $P_1, P_2$ are projective.


%-----------
\exercise
Let $Q_1, Q_2 \in \RMod$. Prove that $Q_1 \oplus Q_2$ is injective if and only if both $Q_1, Q_2$ are injective.


%-----------
\exercise
Prove that the following are equivalent for a ring $R$:
\begin{enumerate}[label={(\roman*)}]
    \item Every $R$-module is projective.
    \item Every $R$-module is injective.
\end{enumerate}


%-----------
\exercise
Let $A$ be a nonzero finite abelian group. 
\bea
\item Prove that $A$ is not a projective $\Z$-module.
\item Prove that $A$ is not an injective $\Z$-module. 
\ee


%-----------
\exercise
\label{ex:infinite-products}
Let $A$ be an $R$-module.
Let $I$ be a nonempty index set, and let $\{B_i\}_{i \in I}$ be a family of $R$-modules. 
Prove the following isomorphisms (of abelian groups):
\bea
\item $\displaystyle \Hom_R(\bigoplus_{i \in I} B_i, A) \cong \prod_{i \in I} \Hom_R(B_i,A)$
\item $\displaystyle \Hom_R(A, \prod_{i \in I} B_i) \cong \prod_{i \in I} \Hom_R(A, B_i)$
\ee
This is yet another way that the functors $\Hom_R(-,A)$ and $\Hom_R(A,-)$ behave differently.


%-----------
\exercise
Let $M$ be a left $R$-module. The \textbf{dual} (module) of $M$ is $\Hom_R(M,R)$, which consists of $R$-module homomorphisms from $M$ to the ground ring $R$. (Note that this is a generalization of the notation of a dual vector space $V^*  = \Hom_{\F}(V, \F)$, where $\F$ is a field.)

Show that the dual of the free $\Z$-module $\bigoplus_{i \in I} \Z$ is \textbf{not} free.  \textit{Hint: Use the previous exercise.}


%-----------
\exercise

Under what conditions on $R$ is ``taking the dual'' an exact functor? 
\textit{Hint: Unpack the definitions. Once the definitions are clear, the solution will be quick.}










\end{document}
