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\begin{document}

\begin{center}
{\bf \Large \underline{Honors Algebra 4, MATH 371 Winter 2010}}\\ 
	{\large Assignment 4}\\Due Wednesday, February 17 at 08:35
\end{center}

\begin{enumerate}
	\item Let $R$ be a commutative ring with $1\neq 0$. 
	\begin{enumerate} 
		\item Prove that the nilradical of $R$ is equal to the intersection of
		the prime ideals of $R$.  Hint: it's easy to show using the definition
		of prime that the nilradical is contained in every prime ideal.  Conversely,
		suppose that $f$ is not nilpotent and consider the set $S$ of ideals
		$I$ of $R$ with the property that ``$n>0\implies f^n\not\in I$."
		Show that $S$ has maximal elements and that any such maximal element must 
		be a prime ideal.  
		
		\item 	Suppose that $R$ is {\em reduced}, {\em i.e.} that the nilradical of $R$ is the zero ideal.
		If $\p$ is a minimal prime ideal of $R$, show that the localization $R_{\p}$
		 has a unique prime ideal and conclude that $R_{\p}$ is a field.
	
		 \item Again supposing $R$ to be reduced, prove that $R$ is isomorphic to a subring
		 of a direct product of fields.
	\end{enumerate}
	
\item Let $R$ be a commutative ring with $1\neq 0$ and let $\varphi:R\rightarrow R$
be a ring homomorphism.  If $R$ is noetherian and $\varphi$ is surjective, 
show that $\varphi$ must be injective too, and hence an isomorphism.
(Hint: Consider the iterates of $\varphi$ and their kernels.)
Can you give a counter-example to this when $R$ is not noetherian?
	
\item As usual, for a prime $p$ we write $\F_p=\Z/p\Z$ for the field with $p$ elements.
\begin{enumerate}
	\item Find all monic irreducible polynomials in $\F_p[X]$ of degree $\le 3$ for $p=2,3,5$.\label{calc}

	\item Prove that for $f\in \F_p[X]$ monic and irreducible, the ideal $(f(X))$ is maximal
	and hence that $\F_p[X]/(f(X))$ is a field.  Show that $\F_p[X]/(f(X))$ has finite 
	cardinality $p^{\deg{f}}$ and use part (\ref{calc}) to explicitly construct finite fields of 
	orders $8,9,25,125$.

	\item Prove that $\F_{7}[X]/(X^2+2)$ and $\F_7[X]/(X^2+X+3)$ are both finite fields of size
	49.  Show that these fields are isomorphic by exhibiting an explicit isomorphism between them.
\end{enumerate}

\item   Let $R$ ba a ring with $1\neq 0$ and $M$ an $R$-module.  
Show that if $N_1\subseteq N_2\subseteq \cdots $
is an ascending chain of submodules of $M$ then $\cup_{i\ge 1} N_i$ is a submodule of $N$.
Show by way of counterexample that modules over a ring need not have maximal proper
submodules (in contrast to the special case of ideals in a ring with $1$).

\item Let $R$ be any commutative ring with $1\neq 0$ and $M$ and $R$-module.  Show that the 
canonical map
$$\Hom_R(R,M)\rightarrow M$$
sending $\varphi$ to $\varphi(1)$ is an isomorphism of $R$-modules.

\item Let $F=\R$ and let $V=\R^3$.  Consider the linear map $\varphi:V\rightarrow V$
given by rotation through an angle of $\pi/2$ about the $z$-axis.  Consider $V$
as an $F[X]$-module by defining
$$(a_nX^n+a_{n-1}X^{n-1}+\cdots + a_1X + a_0)v :=(a_n\varphi^n+a_{n-1}\varphi^{n-1}+\cdots + a_1\varphi + a_0)v,$$
where $\varphi^i$  is the composition of $\varphi$ with itself $i$-times.
\begin{enumerate}
	\item What are the $F[X]$-submodules of $V$?
	\item Show that $V$ is naturally a module over the quotient ring $F[X]/(X^3-X^2+X-1)$.
\end{enumerate}

\item Let $R$ be a ring with $1\neq 0$.  
\begin{enumerate}
	\item For a left ideal $I$ of $R$ and an $R$-module $M$, define
	$$IM:=\left\{  r_1m_1 + r_2m_2 + \cdots + r_k m_k\ :\ r_i\in R,\ m_i\in M,\ k\in \Z_{\ge 0}\right\}.$$
	Show that $IM$ is an $R$-submodule of $M$.

	\item Prove that for any ideal $I$ of $R$ and any positive integer $n$, 
	there is a canonical isomorphism of $R$-modules
	$$R^n/IR^n \simeq R/IR \times R/IR \times \cdots \times R/IR$$
	with $n$-factors in the product on the right.\label{fieldtrick}
	
	\item Suppose now that $R$ is commutative and that $R^n\simeq R^m$ as $R$-modules.
	Show that $m=n$.  Hint: reduce to the case of finite dimensional vector spaces
	over a field by applying (\ref{fieldtrick}) with $I$ a maximal ideal of $R$. 
	
	\item If $R$ is commutative and $A$ is any finite set of cardinality $n$, show that $F(A)\simeq R^n$
	as $R$-modules (Hint: Show that $R^n$ satisfies the same universal mapping property as $F(A)$ 
	and deduce from this that one has maps in both directions whose composition in either order
	must be the identity).  Conclude that the rank of a free module over a commutative ring
	is well-definied if it is finite.
\end{enumerate}

\item Let $R$ be a ring with $1\neq 0$ and $M$ an $R$-module.  We say that $M$ is {\em irreducible}
if $M\neq 0$ and the only submodules of $M$ are $0$ and $M$.  
\begin{enumerate}
	\item Show that $M$ is irreducible if and only if $M$ is a nonzero cyclic $R$-module.
	\item If $R$ is commutative, show that $M$ is irreducible if and only if $M\simeq R/I$
	as $R$-modules for some maximal ideal $I$ of $R$.
	\item Prove Schur's lemma: if $M_1$ and $M_2$ are irreducible $R$-modules then any nonzero
	$R$-module homomorphism $\varphi:M_1\rightarrow M_2$ is an isomorphism.
\end{enumerate}



\end{enumerate}
\end{document}
