# 10-1100/Homework Assignment 5

Jump to: navigation, search

This assignment is due by the end of my office hours (at 12:30) on Thursday December 9, 2010.

### Solve the following questions

Problem 1. Let $M$ be a module over a PID $R$. Assume that $M$ is isomorphic to $R^k\oplus R/\langle a_1\rangle\oplus R/\langle a_2\rangle\oplus\cdots\oplus R/\langle a_l\rangle$, with $a_i$ non-zero non-units and with $a_1\mid a_2\mid\cdots\mid a_l$. Assume also that $M$ is isomorphic to $R^m\oplus R/\langle b_1\rangle\oplus R/\langle b_2\rangle\oplus\cdots\oplus R/\langle b_n\rangle$, with $b_i$ non-zero non-units and with $b_1\mid b_2\mid\cdots\mid b_l$. Prove that $k=m$, that $l=n$, and that $a_i\sim b_i$ for each $i$.

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

1. For each of the $R$-modules $R$, $R/\langle q^t\rangle$, and $R/\langle p^t\rangle$, determine $\ker\hat{p}^s$ and $(R/\langle p\rangle)\otimes\ker\hat{p}^s$.
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 ${\mathbb C}[t]$ modules ${\mathbb C}[t,t^{-1}]$ ("Laurent polynomials in $t$") and ${\mathbb C}$ (here $t$ acts on ${\mathbb C}$ as $0$).

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

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

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

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

1. Suppose that $M$ has rank $n$ and that $x_1,\ldots x_n$ is a maximal set of linearly independent elements of $M$. Show that $\langle x_1,\ldots x_n\rangle$ is isomorphic to $R^n$ and that $M/\langle x_1,\ldots x_n\rangle$ is a torsion module.
2. Converesely show that if $M$ contains a submodule $N$ which is isomorphic to $R^n$ for some $n$, and so that $M/N$ is torsion, then the rank of $M$ is $n$.

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