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\begin{document}
\title{Math 200b Winter 2021 Homework 1}
\date{Due 1/15/2020 by midnight on Gradescope}
\maketitle

1.  Let $R$ be an integral domain.  An $R$-module is called \emph{torsion} 
if for every $m \in M$, there is $0 \neq r \in R$ such that $rm = 0$.
Let $M, N$ and $M_1, M_2, M_3, \dots$ be torsion $R$-modules.
For each question, either prove or give an explicit counterexample.


\medskip

(a).  Must the direct sum $\bigoplus_{n \geq 1} M_n$ be a torsion $R$-module?  

\medskip

(b).  Must the direct product $\prod_{n \geq 1} M_n$ be a torsion $R$-module?

\medskip

(c).  Since $R$ is commutative, $\Hom_R(M, N)$ is again an $R$-module.
Must it be a torsion $R$-module?   Does the answer change if $M$ or $N$ is a finitely generated module?



\bigskip

2.  Let $G$ be a finite group and $F$ a field.  Recall that for a vector space $V$ over $F$, 
the group $\GL(V)$ is the group of all bijective linear transformations from $V$ to itself, with group operation equal to composition.  A \emph{representation} of the group $G$ (over $F$) is a homomorphism of groups $\phi: G \to \GL(V)$ for some vector space $V$.

Let $FG$ be the group algebra of $G$ over $F$.  Show that there is a bijection between $FG$-modules and  representations of $G$ over $F$.

\bigskip

3.  Let $R$ and $S$ be rings.  An abelian group $M$ is an \emph{$(R, S)$-bimodule} if it is both a left $R$-module and a right $S$-module, 
and these two actions are compatible in the sense 
that $(rm)s = r(ms)$ for all $r \in R$, $m \in M$, $s \in S$.  For example, $R$ is an $(R, R)$-bimodule, where $R$ acts on both the left and right by multiplication.  

\medskip

(a).  Suppose that $M$ is an $(R, S)$-bimodule and $N$ is a left $R$-module.  Show that $\Hom_R(M, N)$ is a left $S$-module using the 
action $s \cdot \phi$, where $[s \cdot \phi](m) = \phi(ms)$ for $s \in S$, $\phi \in \Hom_R(M, N)$, $m \in M$.

\medskip

(b).  Suppose that $M$ is a left $R$-module and $N$ is an $(R, T)$-bimodule.  Show that $\Hom_R(M, N)$ is a right $T$-module 
using the action $\phi \cdot t$, where $[\phi \cdot t](m) = \phi(m)t$ 
for $t \in T$, $\phi \in \Hom_R(M, N)$, $m \in M$.  

\medskip

(c).  Suppose that $M$ is an $(R, S)$-bimodule and $N$ is an $(R, T)$-bimodule.  By parts (a) and (b), $\Hom_R(M,N)$ is 
both a left $S$-module and a right $T$-module.  Show that in fact $\Hom_R(M,N)$ is an $(S, T)$-bimodule.  


\bigskip

\newpage

4.  Let $R$ be a commutative ring.   If $I$ is an ideal of $R$ and $M$ is an $R$-module, we write $IM = \{ \sum_{i=1}^n x_i m_i | x_i \in I, m_i \in M \}$,
which is an $R$-submodule of $M$. 

\medskip

(a). Let  $I$ be an ideal of $R$.  Show that if $M$ is an $R$-module, then $M/IM$ is an $R$-module which is also an $R/I$-module via the action
$(r + I) \cdot (m + IM) = rm + IM$.

\medskip

(b).  Recall that two sets $X$ and $Y$ have the same cardinality, written $|X|= |Y|$, if there is a bijective function $f: X \to Y$.  Continue to assume that $R$ is commutative, and suppose that $M$ is a free $R$-module with basis $X$, and $N$ is a free $R$-module with basis $Y$.  Show that $M \cong N$ as $R$-modules if and only if $|X| = |Y|$.  (Hint:  If $M \cong N$, pick any maximal ideal $I$ of $R$ and show that $M/IM \cong N/IN$ is an isomorphism of vector spaces over the field $F = R/I$.   Assume without proof the theorem from linear algebra that any two bases of a vector space have the same cardinality.)

\bigskip

5.  In this problem you will see that the property proved in problem $4(b)$, called \emph{invariance of basis number}, fails for free modules over noncommutative rings in general.

Let $K$ be a field and let $V$ be a countable-dimensional vector space over $K$ with basis 
$v_1, v_2, v_3, \dots$.  Let $R = \operatorname{End}_K(V)$, the ring of all $K$-linear transformations of $V$, where the ring product is function composition as always.   Let $\phi \in R$ be given by 
$\phi(v_i) = v_{i/2}$ for all even $i$, and $\phi(v_i) = 0$ for odd $i$.  Similarly let $\psi \in R$ be given by 
$\psi(v_i) = v_{(i+1)/2}$ for all odd $i$, and $\psi(v_i) = 0$ for all even $i$.  

Show that $R$ is an internal direct sum  $R = R\phi \oplus R \psi$.   Show also 
that $R\phi \cong R \cong R \psi$ as left $R$-modules.  
Conclude that there is an isomorphism of left $R$-modules $R \cong R \oplus R$.   So the free modules of 
rank 1 and 2 over $R$ are isomorphic.

\bigskip

6.  A left $R$-module $M$ is called \emph{simple} or \emph{irreducible} if the only submodules of $M$ 
are $0$ and $M$.

\medskip

(a).   Show that if $R$ is commutative, the simple $R$-modules are exactly the 
cyclic left modules of the form $R/P$ for maximal ideals $P$ of $R$.

\medskip

(b).  Let $M$ be a simple module over any ring $R$.  Show that the ring $\operatorname{End}_R(M)$ is a division ring, that is, that every nonzero element of this ring is a unit.    This result is called \emph{Schur's Lemma}.

\medskip

(c).  The course notes showed that for a field $F$, the vector space of length $n$ column vectors $V  = F^n$ is a left $R = M_n(F)$-module by left matrix multiplication, and $V$ is a simple $R$-module.  What is $\operatorname{End}_R(V)$ in this case?



















 


















\end{document}
