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\begin{document}

\begin{center}
{\bf Math 601 -- Spring 2026 -- Harry Tamvakis}
\end{center}


\begin{center}
{\bf PROBLEM SET 4 -- Due March 5, 2026}
\end{center}


\skipline
\noin
{\bf A1)} Verify that the characteristic polynomial of the matrix
\[
A := \left(\begin{array}{cccc} 1 & 0 & 0 & 0  \\
                        0 & 1 & 0 & 0  \\
                        -2 & -2 & 0 & 1 \\
                        -2 & 0 & -1 & -2 
\end{array}
\right)
\]
is a product of linear factors in $\Q[x]$. Determine the rational 
and Jordan canonical forms for $A$ in $M_n(\Q)$. 


\skipline \noin {\bf A2)}  If $P$ is a projective $R$-module, prove that
there is a free $R$-module $F$ such that $F \cong F\oplus P$. 



\skipline \noin {\bf A3)} Suppose that $P$ and $P'$ are projective
modules over a commutative ring $R$ and that $0 \to K \to P \to M \to
0$ and $0 \to K' \to P' \to M \to 0$ are two short exact sequences of
$R$-modules. Prove that $K\oplus P' \cong K'\oplus P$.  [Hint: If
$\phi:P \to M$ and $\phi':P'\to M$, consider the submodule of
$P\oplus P'$ consisting of those pairs $(x,y)$ such that $\phi(x) =
\phi'(y)$.]


\skipline \noin {\bf A4)} Let $R$ be a commutative ring. In this
problem, an {\em $R$-algebra} means a commutative ring $A$ together
with a homomorphism $\phi:R\to A$. We usually omit $\phi$ from the
notation and write $ra$ for $\phi(r)a$ when $r\in R$ and $a\in
A$. Suppose that for each $R$-algebra $A$ we are given a map of sets
$f_A:A\to A$ such that for every $R$-algebra homomorphism $g:A\to
B$ we have $g\circ f_A = f_B\circ g$. Prove that there
exists a polynomial $p\in R[x]$ such that for every $R$-algebra $A$
and $a\in A$, we have $f_A(a)=p(a)$. [Hint: One way to do this
  uses Yoneda's lemma.]



\skipline
\noin 
{\bf B1)} Problem 24 in Chapter 10.3 of Dummit and Foote (page 358). 
This proves the rather interesting fact that the direct product of 
infinitely many copies of $\Z$ is not a free $\Z$-module.


\skipline
\noin {\bf B2)} Suppose that $R$ is a principal ideal domain. 

\medskip
\noin (a) Let $F$ be a free $R$-module and $E$ a submodule of
$F$. Prove that $E$ is also free. [Hint: We did this in class when $F$
has a finite basis. In general, you will need to use Zorn's Lemma. One
suggestion is to let $B$ be a basis of $F$ and consider the set of
pairs $(A,L)$ such that $A\subset B$ and $R\langle A \rangle\cap E$ is
free with basis $L$, ordered by inclusion in the natural way.]

\medskip
\noin
(b) Show that any projective $R$-module is free. It follows from 
this and Problem B1 that the direct product of projective modules 
need not be projective.



\skipline
\noin
{\bf B3)} Let $R$ be a commutative ring and $P$ be a finitely generated
projective $R$-module.

\medskip
\noin
(a) Write $P^*$ for the dual module $\Hom_R(P,R)$. Prove that the natural 
map $P\to P^{**}$ is an isomorphism of $R$-modules.

\medskip
\noin
(b) Let $\End(P) = \Hom_R(P,P)$. Show that there is a 
natural isomorphism $P^*\otimes P \cong \End(P)$ [Hint: one way to 
do this uses part (a)]. The $R$-bilinear map 
$P^*\times P \to R$ sending $(f,x)$ to $f(x)$ induces an $R$-linear map
$\Tr: P^*\otimes P \to R$, and hence a map 
$\Tr: \End(P)\to R$. 
Show that if $P=R^n$ is free of rank $n$, then $\Tr$ can be identified
with the usual trace of matrices in $M_n(R)$.

\medskip
\noin
(c) Suppose now that $\mu : \End(P) \to R$ is any $R$-linear map. Prove that 
there exists a unique element $g\in \End(P)$ such that 
$\mu(f) = \Tr(fg)$, for all $f\in \End(P)$.



\begin{center}
{\bf \underline{Extra Credit Problem}}
\end{center}


\skipline
\noin
{\bf C1)} (a) If $N$ is a nilpotent $n\times n$ matrix over $\C$, use
the binomial series for $\root\of{1+x}$ to obtain a formula for the
square root of $I+N$ (that is, a matrix $A$ with $A^2=I+N$).

\medskip
\noin
(b) Use the Jordan form to prove that every invertible complex
$n\times n$ matrix has a square root.

\medskip
\noin
(c) Suppose $J=J_k(0)$ is a Jordan block of size $k$ for the eigenvalue
0. Find the Jordan canonical form for $J^2$. Use this and part (a) to 
determine necessary and sufficient conditions for a matrix $A\in
M_n(\C)$ to have a square root.

\medskip
\noin
(d) Let $F$ be a field of characteristic 2 (such as
$\F_2=\Z/2\Z$). Determine necessary and sufficient conditions for 
a matrix $A\in M_n(F)$ to have a square root (note that you cannot
use the idea in part (a) here.)





\end{document}
