Welcome to Math 1100! (additions to this web site no longer count towards good deed points)

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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, JordanHolder 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, OrbitStabilizer Thm, Class Equation.

4

Sep 29

Monday  Cauchy's Thm, Sylow 1; Thursday  Sylow 2.

5

Oct 6

Monday  Sylow 3, semidirect 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, CayleyHamilton Thm, ideals, iso thm 1; Thursday  iso thms 24, integral domains, maximal ideals, One Theorem, Three Corollaries, Five Weeks

9

Nov 3

Monday  prime ideals, primes & irreducibles, UFD's, Euc.Domain$\Rightarrow$PID, Thursday  Noetherian rings, PID$\Rightarrow$UFD, Euclidean Algorithm, modules: defn & examples, HW4, HW 4 Solutions

10

Nov 10

Monday  R is a PID iff R has a DH norm, Rmodules, 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

MondayTuesday 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

EndofCourse 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

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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 $M$ 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 $a_{i}$ nonzero nonunits 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 $b_{i}$ nonzero nonunits 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.
 For each of the $R$modules $R$, $R/\langle q^{t}\rangle$, and $R/\langle p^{t}\rangle$, determine $\ker {\hat {p}}^{s}$ and $(R/\langle p\rangle )\otimes \ker {\hat {p}}^{s}$.
 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 $R$linearlyindependent 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 nonzero $r\in R$ 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 $R$.
 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.
 Conversely 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 $R^{k}\oplus \bigoplus R/\langle p_{i}^{s_{i}}\rangle$.