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\begin{document}
\title{Math 200b Winter 2021 Homework 4}
\date{Due 2/5/2020 by midnight on Gradescope}
\maketitle


1.   Let $V$ be a vector space over a field $F$, with 
$v_1, v_2 \in V$ be linearly independent over $F$.  Show 
that $v_1 \otimes v_1 + v_2 \otimes v_2 \in V \otimes_F V$ is an element which is not equal to any pure tensor $u \otimes w$ 
with $u, w \in V$.

\bigskip

2.  Let $R$ be a commutative ring.

(a).  Show that for any ideals $I$ and $J$ of $R$, that 
\[
R/I \otimes_R R/J \cong R/(I +J)
\]
as $R$-modules.  

(b).  Suppose that $R$ is a PID and $M$ and $N$ are finitely generated torsion $R$-modules.  Show that $M \otimes_R N$ is also finitely generated as an $R$-module.  Describe 
the elementary divisors of $M \otimes_R N$ in terms of the elementary divisors of $M$ and $N$.

\bigskip


3.  Let $F \subseteq K$ be an inclusion of fields.  Then $K$ is an $F$-algebra in a natural way.  Since $K$ and $F[x]$ are $F$-algebras, 
$K \otimes_F F[x]$ is also an $F$-algebra.

(a).  Prove that $K \otimes_F F[x]$ is actually a $K$-algebra and that there is a $K$-algebra isomorphism $K \otimes F[x] \cong K[x]$.

(b).  Consider the ring $R = F[x]/(g(x))$ for some $g \in F[x]$.  Prove that $K \otimes_F R \cong K[x]/(g(x))$ as $K$-algebras.

(c).  Show that $\mb{C} \otimes_{\mb{R}} \mb{C} \cong \mb{C} \times \mb{C}$ as $\mb{C}$-algebras.  (Hint:  $\mb{C} \cong \mb{R}[x]/(x^2 + 1)$.)


\bigskip

4.  Let $R$ be an integral domain with field of fractions $F$.  
Let $M$ be a finitely generated $R$-module.  Show that $F \otimes_R M$ is a finite-dimensional $F$-vector space.  We define  the  \emph{rank} of $M$ to be 
$\operatorname{rk}(M) = \dim_F (F \otimes_R M)$. 

(a).  If $M$ is a torsion $R$-module, then $F \otimes_R M = 0$.

(b).  Show that if  $\operatorname{rk}(M) = r$, then $r$ is equal to the largest natural 
number such that $M$ contains a submodule $F$ which is free of rank $r$.

(c).  If $R$ is a PID, then $\operatorname{rk}(M)$ is equal to the rank of $M$ as 
we defined it earlier, as the unique number $r$ such that $M \cong R^r \oplus T$ where 
$T$ is a torsion module.







\end{document}
