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

In Preparation

The information below is preliminary and cannot be trusted! (v)

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

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

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

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

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

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

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

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

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