# 14-1100/Homework Assignment 5

This assignment is extended from class time on Wednesday, December 3, 2014 (a "virtual Monday" and the last day of the semester) to the end of Monday, December 8 in Dror's mailbox.

### Solve the following questions

Problem 1. Let ${\displaystyle M}$ be a module over a PID ${\displaystyle R}$. Assume that ${\displaystyle M}$ is isomorphic to ${\displaystyle R^{k}\oplus R/\langle a_{1}\rangle \oplus R/\langle a_{2}\rangle \oplus \cdots \oplus R/\langle a_{l}\rangle }$, with ${\displaystyle a_{i}}$ non-zero non-units and with ${\displaystyle a_{1}\mid a_{2}\mid \cdots \mid a_{l}}$. Assume also that ${\displaystyle M}$ is isomorphic to ${\displaystyle R^{m}\oplus R/\langle b_{1}\rangle \oplus R/\langle b_{2}\rangle \oplus \cdots \oplus R/\langle b_{n}\rangle }$, with ${\displaystyle b_{i}}$ non-zero non-units and with ${\displaystyle b_{1}\mid b_{2}\mid \cdots \mid b_{l}}$. Prove that ${\displaystyle k=m}$, that ${\displaystyle l=n}$, and that ${\displaystyle a_{i}\sim b_{i}}$ for each ${\displaystyle i}$.

Problem 2. Let ${\displaystyle q}$ and ${\displaystyle p}$ be primes in a PID ${\displaystyle R}$ such that ${\displaystyle p\not \sim q}$, let ${\displaystyle {\hat {p}}}$ denote the operation of "multiplication by ${\displaystyle p}$", acting on any ${\displaystyle R}$-module ${\displaystyle M}$, and let ${\displaystyle s}$ and ${\displaystyle t}$ be positive integers.

1. For each of the ${\displaystyle R}$-modules ${\displaystyle R}$, ${\displaystyle R/\langle q^{t}\rangle }$, and ${\displaystyle R/\langle p^{t}\rangle }$, determine ${\displaystyle \ker {\hat {p}}^{s}}$ and ${\displaystyle (R/\langle p\rangle )\otimes \ker {\hat {p}}^{s}}$.
2. Explain why this approach for proving the uniqueness in the structure theorem for finitely generated modules fails.

Problem 3. (comprehensive exam, 2009) Find the tensor product of the ${\displaystyle {\mathbb {C} }[t]}$ modules ${\displaystyle {\mathbb {C} }[t,t^{-1}]}$ ("Laurent polynomials in ${\displaystyle t}$") and ${\displaystyle {\mathbb {C} }}$ (here ${\displaystyle t}$ acts on ${\displaystyle {\mathbb {C} }}$ as ${\displaystyle 0}$).

Problem 4. (from Selick) Show that if ${\displaystyle R}$ is a PID and ${\displaystyle S}$ is a multiplicative subset of ${\displaystyle R}$ then ${\displaystyle S^{-1}R}$ is also a PID.

Definition. The "rank" of a module ${\displaystyle M}$ over a (commutative) domain ${\displaystyle R}$ is the maximal number of ${\displaystyle R}$-linearly-independent elements of ${\displaystyle M}$. (Linear dependence and independence is defined as in vector spaces).

Definition. An element ${\displaystyle m}$ of a module ${\displaystyle M}$ over a commutative domain ${\displaystyle R}$ is called a "torsion element" if there is a non-zero ${\displaystyle r\in R}$ such that ${\displaystyle rm=0}$. Let ${\displaystyle {\mbox{Tor }}M}$ denote the set of all torsion elements of ${\displaystyle M}$. (Check that ${\displaystyle {\mbox{Tor }}M}$ is always a submodule of ${\displaystyle M}$, but don't bother writing this up). A module ${\displaystyle M}$ is called a "torsion module" if ${\displaystyle M={\mbox{Tor }}M}$.

Problem 5. (Dummit and Foote, page 468) Let ${\displaystyle M}$ be a module over a commutative domain ${\displaystyle R}$.

1. Suppose that ${\displaystyle M}$ has rank ${\displaystyle n}$ and that ${\displaystyle x_{1},\ldots x_{n}}$ is a maximal set of linearly independent elements of ${\displaystyle M}$. Show that ${\displaystyle \langle x_{1},\ldots x_{n}\rangle }$ is isomorphic to ${\displaystyle R^{n}}$ and that ${\displaystyle M/\langle x_{1},\ldots x_{n}\rangle }$ is a torsion module.
2. Conversely show that if ${\displaystyle M}$ contains a submodule ${\displaystyle N}$ which is isomorphic to ${\displaystyle R^{n}}$ for some ${\displaystyle n}$, and so that ${\displaystyle M/N}$ is torsion, then the rank of ${\displaystyle M}$ is ${\displaystyle n}$.

Problem 6. (see also Dummit and Foote, page 469) Show that the ideal ${\displaystyle \langle 2,x\rangle }$ in ${\displaystyle R={\mathbb {Z} }[x]}$, regarded as a module over ${\displaystyle R}$, is finitely generated but cannot be written in the form ${\displaystyle R^{k}\oplus \bigoplus R/\langle p_{i}^{s_{i}}\rangle }$.