14-1100/Homework Assignment 5

DBN: Classes: 2014-15: 14-1100/Navigation Welcome to Math 1100! (additions to this web site no longer count towards good deed points) # Week of... Notes and Links 1 Sep 8 About This Class; Monday - Non Commutative Gaussian Elimination; Thursday - the category of groups, automorphisms and conjugations, images and kernels. 2 Sep 15 Monday - coset spaces, isomorphism theorems; Thursday - simple groups, Jordan-Holder decomposition series. 3 Sep 22 Monday - alternating groups, group actions, The Simplicity of the Alternating Groups, HW1, HW 1 Solutions, Class Photo; Thursday - group actions, Orbit-Stabilizer Thm, Class Equation. 4 Sep 29 Monday - Cauchy's Thm, Sylow 1; Thursday - Sylow 2. 5 Oct 6 Monday - Sylow 3, semi-direct products, braids; HW2; HW 2 Solutions; Thursday - braids, groups of order 12, Braids 6 Oct 13 No class Monday (Thanksgiving); Thursday - groups of order 12 cont'd. 7 Oct 20 Term Test; Term Test Solutions on Monday, HW3; HW 3 Solutions; Thursday - solvable groups, rings: defn's & examples. 8 Oct 27 Monday - functors, Cayley-Hamilton Thm, ideals, iso thm 1; Thursday - iso thms 2-4, integral domains, maximal ideals, One Theorem, Three Corollaries, Five Weeks 9 Nov 3 Monday - prime ideals, primes & irreducibles, UFD's, Euc.Domain[math]\displaystyle{ \Rightarrow }[/math]PID, Thursday - Noetherian rings, PID[math]\displaystyle{ \Rightarrow }[/math]UFD, Euclidean Algorithm, modules: defn & examples, HW4, HW 4 Solutions 10 Nov 10 Monday - R is a PID iff R has a D-H norm, R-modules, direct sums, every f.g. module is given by a presentation matrix, Thursday - row & column reductions plus, existence part of Thm 1 in 1t3c5w handout. 11 Nov 17 Monday-Tuesday is UofT's Fall Break, HW5, Thursday - 1t3c5w handout cont'd, JCF Tricks & Programs handout 12 Nov 24 Monday - JCF Tricks & Programs cont'd, tensor products, Thursday - tensor products cont'd 13 Dec 1 End-of-Course Schedule; Monday - tensor products finale, extension/reduction of scalars, uniqueness part of Thm 1 in 1t3c5w, localization & fields of fractions; Wednesday is a "makeup Monday"!; Notes for Studying for the Final Exam Glossary of terms F Dec 15 The Final Exam Register of Good Deeds Add your name / see who's in! See Non Commutative Gaussian Elimination

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 [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], that [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. Conversely 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].