11-1100/Homework Assignment 5

In Preparation

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

DBN: Classes: 2011-12: 11-1100/Navigation # Week of... Notes and Links Additions to the MAT 1100 web site no longer count towards good deed points 1 Sep 12 About This Class, Tuesday - Non Commutative Gaussian Elimination, Thursday - NCGE completed, the category of groups, images and kernels. 2 Sep 19 I'll be in Strasbourg, class was be taught by Paul Selick. See my summary. 3 Sep 26 Class Photo, HW1, HW1 Submissions, HW 1 Solutions 4 Oct 3 The Simplicity of the Alternating Groups 5 Oct 10 HW2, HW 2 Solutions 6 Oct 17 Groups of Order 60 and 84 7 Oct 24 Extra office hours: Monday 10:30-12:30 (Dror), 5PM-7PM @ Huron 1028 (Stephen). Term Test on Tuesday. Summary for the midterm exam. Term Test - Sample Solutions 8 Oct 31 HW3, HW 3 Solutions 9 Nov 7 Monday-Tuesday is November Break, One Theorem, Two Corollaries, Four Weeks 10 Nov 14 HW4, HW 4 Solutions 11 Nov 21 12 Nov 28 HW5 and last week's schedule 13 Dec 5 Tuesday - the Jordan form; UofT Fall Semester ends Wednesday; our Final Exam took place on Friday Register of Good Deeds Add your name / see who's in! See Non Commutative Gaussian Elimination

Last Week's Schedule

Warning. This schedule is subject to changes. Recheck this web site the day before any activity.

Tuesday December 6	10-12	Last Class
Wednesday December 7	12-2	Dror's office hours, Bahen 6178.
	2PM	HW5 "early bird" due date. If you submit HW5 by this time, it will be marked by noon of the following day.
Thursday December 8	10:30-12:30	Dror's office hours, Bahen 6178.
	Noon	HW5 is due in Dror's office, to be graded after the final. Also, at this time "early bird" marked HW5 can be collected at Dror's office.
	3-5	Stephen Morgan's office hours, at Huron 1028.
Friday December 9	10-1	The Final Exam.

Solve the following questions

Problem 1. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} be a module over a PID $R$. Assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^k\oplus R/\langle a_1\rangle\oplus R/\langle a_2\rangle\oplus\cdots\oplus R/\langle a_l\rangle} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i} non-zero non-units and with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1\mid a_2\mid\cdots\mid a_l} . Assume also that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^m\oplus R/\langle b_1\rangle\oplus R/\langle b_2\rangle\oplus\cdots\oplus R/\langle b_n\rangle} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_i} non-zero non-units and with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a multiplicative subset of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{-1}R} is also a PID.

Definition. The "rank" of a module Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} over a (commutative) domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is the maximal number of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -linearly-independent elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . (Linear dependence and independence is defined as in vector spaces).

Definition. An element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} of a module Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} over a commutative domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is called a "torsion element" if there is a non-zero Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\in R} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle rm=0} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Tor }M} denote the set of all torsion elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . (Check that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Tor }M} is always a submodule of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , but don't bother writing this up). A module Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is called a "torsion module" if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=\mbox{Tor }M} .

Problem 5. (Dummit and Foote, page 468) Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} be a module over a commutative domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .

1. Suppose that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} has rank Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} and that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1,\ldots x_n} is a maximal set of linearly independent elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} . Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x_1,\ldots x_n\rangle} is isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^n} and that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M/\langle x_1,\ldots x_n\rangle} is a torsion module.
2. Converesely show that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} contains a submodule Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} which is isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^n} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , and so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M/N} is torsion, then the rank of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

Problem 6. (see also Dummit and Foote, page 469) Show that the ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle 2,x\rangle} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R={\mathbb Z}[x]} , regarded as a module over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , is finitely generated but cannot be written in the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^k\oplus\bigoplus R/\langle p_i^{s_i}\rangle} .