
\documentclass{amsart}

\title{Math studies algebra Spring 2018: Homework 4}

\usepackage{amssymb} 

\usepackage{tikz-cd}

\author{JC}

\begin{document}

\maketitle

   This homework is due by class time on Monday 19 February. It must be typeset
   (preferably in\ \LaTeX) and submitted as a PDF file on the Canvas site,
   with a filename of the form \begin{verbatim} andrewID_alg_homeworknumber.pdf \end{verbatim}   
   For each minute that it is late, the grade will be reduced by 10 percent. 

\medskip

\begin{enumerate}


\item   Let $A$ be an $n \times n$ matrix with integer entries. Let $G$ be the subgroup of ${\mathbb Z}^n$ generated by the columns 
     of $A$.

\begin{enumerate}

\item   Prove that the rank (in the sense of ${\mathbb Z}$-modules) of $G$ is equal to the rank (in the sense of matrices with real entries)
     of the matrix $A$.

\item   Prove that ${\mathbb Z}^n/G$ is finite if and only if $A$ is non-singular, and in this case
        $\vert {\mathbb Z}^n/G \vert = \vert \det(A) \vert$.


\end{enumerate}  


\item  Let $R$ be a commutative ring with $1$, let $I$ be an ideal of $R$ and let $M$ be an $R$-module. Let $I M$ be the
   subset of $M$ consisting of elements of the form $\sum_{i=1}^n r_i m_i$ for $r_i \in I$, $m_i \in M$.

\begin{enumerate}

\item  Prove that $I M$ is a submodule of $M$.

\item  Prove that if we attempt to define a scalar multiplication map from $R/I \times M/IM$ to $M/IM$ by
    $(r + I)(m + IM) = rm + IM$, then we get a well-defined map which makes $M/IM$ into an $R/I$-module. 

\item Prove that if $R$ is not the zero ring and $R^m$ is isomorphic to $R^n$ as an $R$-module for positive integers
   $m$ and $n$, then $m  = n$. Hint: $R$ has at least one maximal ideal. 


\end{enumerate}


\item  Let $R$ be a commutative ring with $1$ and let $M$ be an $R$-module such that $M \oplus N$ is free for some $R$-module $N$.
 Prove that if $\alpha: N_1 \rightarrow N_2$ is any surjective   $R$-linear map, then $Hom(M, \alpha) : Hom(M, N_1) \rightarrow Hom(M, N_2)$ is also surjective.  
 Hint: Start with the easier case where $M$ itself is free. 

\item  Fill in the steps in the following outline of a proof that if $R$ is a PID, $N$ is a free $R$-module (possibly of infinite rank) 
  and $M \le N$ then $M$ is also free. You should add a  proof for each claim (many of them will be short).  


\begin{enumerate}

\item    Claim 1: $N$ has a basis, say $\{ e_i : i \in I \}$ for some index set $I$.

\item    Claim 2: The set $I$ has a well-ordering $<_I$.  

\item    Claim 3: There is a linear map $p_i: N \rightarrow R$ such that $p_i(e_j) = 1$ for $i = j$ and $0$ otherwise.

\item    Claim 4: If $N_j = span \{ e_i : i \le_I j \}$, then $N_j \cap M \le N$.

\item    Claim 5: $p_j[N_j \cap M]$ is an ideal of $R$.

\item    Claim 6: $p_j[N_j \cap M] = R a_j$ for some $a_j \in R$.

\item    Claim 7: There is $m_j \in N_j \cap M$ such that $p_j(m_j) = a_j$. 
    
\item    Claim 8: $\{ m_j : a_j \neq 0 \}$ is linearly independent. Hint: use the $p_j$'s
         and the fact that  any finite subset of $I$ has a largest element.

\item    Claim 9: $\{ m_j : a_j \neq 0 \}$ spans $M$. Hint: Given nonzero $n \in N$, define $i(n)$ to be
   the largest $i \in I$ such that $e_i$ appears with nonzero coefficient in the expansion of $n$.
   If the given set is not spanning, start by choosing $m \in M$ outside the span with $i(m)$ minimal.


\end{enumerate}  


\item An $n \times n$ real matrix $A$ is said to be {\em skew-symmetric} if $A + A^T = 0$, and
     {\em orthogonal} if $A A^T = I$.  

\begin{enumerate}

\item Let $A(t)$ be a differentiable matrix-valued function (taking values in the set
   of $n \times n$ real matrices) such that $A(t)$ is orthogonal for all $t$ and
   $A(0) = I$. Prove that $A'(0)$ is skew-symmetric. 

\item  Prove that if $B$ is skew-symmetric and $A(t) = \exp(B t)$, then
   $A(t)$ is orthogonal for all $t$ and $A'(0) = B$. 




\end{enumerate}  

\end{enumerate}  

     
\end{document} 
