|Welcome to Math 1100!|
(additions to this web site no longer count towards good deed points)
||Notes and Links
||About This Class; Monday - Non Commutative Gaussian Elimination; Thursday - the category of groups, automorphisms and conjugations, images and kernels.
||Monday - coset spaces, isomorphism theorems; Thursday - simple groups, Jordan-Holder decomposition series.
||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.
||Monday - Cauchy's Thm, Sylow 1; Thursday - Sylow 2.
||Monday - Sylow 3, semi-direct products, braids; HW2; HW 2 Solutions; Thursday - braids, groups of order 12, Braids
||No class Monday (Thanksgiving); Thursday - groups of order 12 cont'd.
||Term Test; Term Test Solutions on Monday, HW3; HW 3 Solutions; Thursday - solvable groups, rings: defn's & examples.
||Monday - functors, Cayley-Hamilton Thm, ideals, iso thm 1; Thursday - iso thms 2-4, integral domains, maximal ideals, One Theorem, Three Corollaries, Five Weeks
||Monday - prime ideals, primes & irreducibles, UFD's, Euc.DomainPID, Thursday - Noetherian rings, PIDUFD, Euclidean Algorithm, modules: defn & examples, HW4, HW 4 Solutions
||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.
||Monday-Tuesday is UofT's Fall Break, HW5, Thursday - 1t3c5w handout cont'd, JCF Tricks & Programs handout
||Monday - JCF Tricks & Programs cont'd, tensor products, Thursday - tensor products cont'd
||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
||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 be a module over a PID . Assume that is isomorphic to , with non-zero non-units and with . Assume also that is isomorphic to , with non-zero non-units and with . Prove that , that , and that for each .
Problem 2. Let and be primes in a PID such that , let denote the operation of "multiplication by ", acting on any -module , and let and be positive integers.
- For each of the -modules , , and , determine and .
- 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 modules ("Laurent polynomials in ") and (here acts on as ).
Problem 4. (from Selick) Show that if is a PID and is a multiplicative subset of then is also a PID.
Definition. The "rank" of a module over a (commutative) domain is the maximal number of -linearly-independent elements of . (Linear dependence and independence is defined as in vector spaces).
Definition. An element of a module over a commutative domain is called a "torsion element" if there is a non-zero such that . Let denote the set of all torsion elements of . (Check that is always a submodule of , but don't bother writing this up). A module is called a "torsion module" if .
Problem 5. (Dummit and Foote, page 468) Let be a module over a commutative domain .
- Suppose that has rank and that is a maximal set of linearly independent elements of . Show that is isomorphic to and that is a torsion module.
- Conversely show that if contains a submodule which is isomorphic to for some , and so that is torsion, then the rank of is .
Problem 6. (see also Dummit and Foote, page 469) Show that the ideal in , regarded as a module over , is finitely generated but cannot be written in the form .