10-1100/Homework Assignment 5

DBN: Classes: 2010-11: 10-1100/Navigation Additions to the MAT 1100 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 13 About This Class, Tuesday - Non Commutative Gaussian Elimination, Thursday 2 Sep 20 Tuesday - Homomorphisms and Normal Groups, Thursday - Isomorphism Theorems 3 Sep 27 Class Photo, HW1, HW1 solution 4 Oct 4 5 Oct 11 HW2, HW2 solution 6 Oct 18 7 Oct 25 Term Test 8 Nov 1 HW3, HW3 solution 9 Nov 8 Monday-Tuesday is Fall Break, One Theorem, Two Corollaries, Four Weeks 10 Nov 15 HW4, HW4 solution 11 Nov 22 12 Nov 29 HW5, HW5 solution 13 Dec 6 Boxing Day Handout, see also December 2010 Schedule. Some Class Notes F Dec 13 Final exam, Tuesday December 14 10-1, Bahen 6183 Register of Good Deeds Add your name / see who's in! See Non Commutative Gaussian Elimination

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 $R^{k}\oplus \bigoplus R/\langle p_{i}^{s_{i}}\rangle$.