\magnification = \magstep1
\def\im{{\rm im}}
\centerline{MATH 400 -- Directed Readings in Module and Representation Theory}
\centerline{Midterm Problem Set -- Due Friday, March 19 (no later than 5:00pm)}
\bigskip
\noindent
{\it Directions and Groundrules:}  
You may consult Dummit and Foote for definitions, background, etc.
Do not consult other books, other faculty members, etc.
\bigskip
\noindent 
I.  Let $R$ be a commutative ring with $1 \ne 0$.  
Recall that an ideal $M \ne R$ in 
$R$ is said to be {\it maximal} if whenever $I$ is an ideal of 
$R$ such that $M \subseteq I \subseteq R$, then either $I = M$ or
$I = R$.  
\item{A)}  Show that there exist maximal ideals in $R$.  
\item{B)}  Show that if $M$ is an ideal in $R$, then $M$ is maximal if and only if $R/M$ is a field. 
\item{C)}  Let $S = R \oplus \cdots \oplus R$ ($n$ terms), 
as a module over $R$. If $M$ is any ideal in $R$, 
show that $N = M \oplus \cdots \oplus M$ is an submodule in $S$
and identify the quotient $S/N$ up to isomorphism.   
\item{D)}  In this part and the next, $m,n \in {\bf N}$.  
Show that if $F$ is a field, then $F^n \simeq F^m$
as vector spaces over $F$ if and only if $n = m$.  
\item{E)}  Show that $R^n \simeq R^m$
(isomorphic as $R$-modules) if and only if $n = m$.  
\bigskip
\noindent
II.  A {\it local ring} is a commutative ring with $1$ which 
has a unique maximal ideal.  
\item{A)}  Show that ${\bf Z}/p^k{\bf Z}$ is a local ring
for any prime $p$ and $k \ge 1$, by identifying the unique
maximal ideal. 
\item{B)}  Let $F$ be a field and let 
$F[x]_{(x)}$ be the subring of the field of rational functions $F(x)$
consisting of ``fractions'' $f(x)/g(x)$ such that $g(x) \notin (x)$
(that is such that $g(0) \ne 0$).  (You can think of $F[x]_{(x)}$ as
the subring of the field $F(x)$ of rational functions for which 
evaluation at $x = 0$ makes sense.)
Show that $F[x]_{(x)}$ is a local ring and identify the maximal ideal. 
\item{C)}  An important fact about modules over local rings is
the following:  Show that if $M$ is a finitely generated module over
a local ring $R$ with unique maximal ideal $I$ such that $IM = M$, then
$M = \{0\}$.  
\item{D)}  Deduce that if $M$ is a finitely generated module over
the local ring $R$ with unique maximal ideal $I$
and $N$ is a submodule such that $M = N + IM$, then $N = M$.
\item{E)}  Deduce that if $M$ is a finitely generated module
over a local ring $R$ with maximal ideal $I$ and $x_1 + IM,\ldots,x_s+IM$
generate $M/IM$, then $x_1,\ldots, x_s$ generate $M$.     
\bigskip
\noindent
III.  Let $I$ be a nonempty index set and for each $i \in I$, let
$N_i$ be a submodule of the $R$-module $M$.  Show that the following
are equivalent:
\item{(i)}  The submodule of $M$ generated by the $N_i$ is isomorphic
to the direct sum $\oplus_{i\in I} N_i$.
\item{(ii)}  If ${i_1,\ldots,i_k}$ is any finite subset of $I$, then
$N_{i_1} \cap (N_{i_2} + \cdots + N_{i_k}) = \{0\}$.  
\item{(iii)} If ${i_1,\ldots,i_k}$ is any finite subset of $I$, then
$N_{i_1} + \cdots + N_{i_k}) \simeq N_{i_1} \oplus \cdots \oplus N_{i_k}$.
\item{(iv)}  For any element $x$ of the submodule generated by the
$N_i$, there are unique $a_i \in N_i$, all but finitely many of which are 
zero, and such that $x = \sum_{i\in I} a_i$ (note this is a finite sum).  
\bigskip
\noindent
IV.  Let $R$ be a commutative ring with $1$.
\item{A)}  Prove that if $M,N$ are free $R$-modules, then 
$M \otimes_R N$ is free.  
\item{B)}  Deduce that if $M,N$ are projective,
then $M \otimes_R N$ is projective.     
\item{C)}  Show that the polynomial ring 
$R[x]$ is a flat module over $R$.
\bigskip
\noindent
V.  Let $A$ be an $n\times n$ matrix with entries in $F$
that satisfies $A^2 = A$.
\item{A)}  If $n = 5$, find all possible rational canonical
forms of $A$.  
\item{B)}  For all $n$, show that $F^n = \ker(A) \oplus \im(A)$.  
Deduce that $A$ is always diagonalizable over $F$. 
\bigskip
\noindent
VI.  Let $V,W$ be finite-dimensional vector spaces over an algebraically closed
field $F$.  Let $A,B$ be linear mappings $A : V \to V$ and $B : W \to W$.
Suppose the characteristic polynomials of $A$ and $B$ factor as
$c(A) = \prod_i (x - \alpha_i)^{n_i}$ and $c(B) = \prod_j (x - \beta_j)^{m_j}$
(where the $\alpha_i$ and $\beta_j$ are distinct).
\item{A)}  What is the characteristic polynomial of the mapping 
$A \oplus B : V \oplus W \to V\oplus W$?  
\item{B)}  If $A$ and $B$ are diagonalizable, 
what is the characteristic polynomial of the mapping 
$A \otimes_F B : V \otimes_F W \to V\otimes_F W$?  
\bye
