Drorbn - User contributions [en]

14-1100/Classnotes for Monday November 10

2014-12-10T18:16:52Z

Anai:

{{14-1100/Navigation}}

Here are some handwritten notes about direct sums in a bit more detail:
{|
|[[File:MAT1100 Lec_17, 14-11-10M - direct sums, small p1.pdf]]
|[[File:MAT1100 Lec 17, 14-11-10M - direct sums, small p2.pdf]]
|}
Handwritten notes about presentation matrices for finitely generated modules in a bit more detail:
''Apologies''! I had found a few mistakes my notes: there are a few places where I'd flipped <math>R^n</math> with X, and other similar errors.

Here are some jpg's with corrections, although there may still be other mistakes...
{|
|[[File: 14-1100 Lec 17, 14-11-10M - module gen by a matrix, small p1.jpg|200px|thumb|left|page 1]]
|[[File: 14-1100 Lec 17, 14-11-10M - module gen by a matrix, small p2.jpg|200px|thumb|left|page 2]]
|}
{|
|[[File: 14-1100 Lec 17, 14-11-10M - module gen by a matrix, small p3.jpg|200px|thumb|left|page 3]]
|}

{|
|[[File:14-1100 Lec 17 board 1 flip.JPG|200px|thumb|left|Lec 17 board 1]]
|[[File:14-1100 Lec 17 board 2 flip.JPG|200px|thumb|left|Lec 17 board 2]]
|}
{|
|[[File:14-1100 Lec 17 board 3 flip.JPG|200px|thumb|left|Lec 17 board 3]]
|[[File:14-1100 Lec 17 board 4 flip.JPG|200px|thumb|left|Lec 17 board 4]]
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{|
|[[File:14-1100 Lec 17 board 5 flip.JPG|200px|thumb|left|Lec 17 board 5]]
|[[File:14-1100 Lec 17 board 6 flip.JPG|200px|thumb|left|Lec 17 board 6]]
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{|
|[[File:14-1100 Lec 17 board 7 flip.JPG|200px|thumb|left|Lec 17 board 7]]
|[[File:14-1100 Lec 17 board 8 flip.JPG|200px|thumb|left|Lec 17 board 8]]
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{|
|[[File:14-1100 Lec 17 board 9 flip.JPG|200px|thumb|left|Lec 17 board 9]]
|[[File:14-1100 Lec 17 board 10 flip.JPG|200px|thumb|left|Lec 17 board 10]]
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{|
|[[File:14-1100 Lec 17 board 11 flip.JPG|200px|thumb|left|Lec 17 board 11]]
|[[File:14-1100 Lec 17 board 12 flip.JPG|200px|thumb|left|Lec 17 board 12]]
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{|
|[[File:14-1100 Lec 17 board 13 flip.JPG|200px|thumb|left|Lec 17 board 13]]
|[[File:14-1100 Lec 17 board 14 flip.JPG|200px|thumb|left|Lec 17 board 14]]
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{|
|[[File:14-1100 Lec 17 board 15 flip.JPG|200px|thumb|left|Lec 17 board 15]]
|[[File:14-1100 Lec 17 board 16 flip.JPG|200px|thumb|left|Lec 17 board 16]]
|}
{|
|[[File:14-1100 Lec 17 board 17 flip.JPG|200px|thumb|left|Lec 17 board 17]]
|}

'''Here are some handwritten notes:'''

{|
|[[File:14-1100 Lec 17 page 1.JPG|200px|thumb|left|Lec 15 page 1]]
|[[File:14-1100 Lec 17 page 2.JPG|200px|thumb|left|Lec 15 page 2]]
|[[File:14-1100 Lec 17 page 3.JPG|200px|thumb|left|Lec 15 page 3]]
|[[File:14-1100 Lec 17 page 4.JPG|200px|thumb|left|Lec 15 page 4]]
|}
{|
|[[File:14-1100 Lec 17 page 5.JPG|200px|thumb|left|Lec 15 page 5]]
|}

File:14-1100 Lec 17, 14-11-10M - module gen by a matrix, small p3.jpg

2014-12-10T18:11:35Z

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File:14-1100 Lec 17, 14-11-10M - module gen by a matrix, small p2.jpg

2014-12-10T18:11:11Z

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File:14-1100 Lec 17, 14-11-10M - module gen by a matrix, small p1.jpg

2014-12-10T18:10:36Z

Anai:

14-1100/Classnotes for Monday November 10

2014-12-10T03:09:56Z

Anai:

{{14-1100/Navigation}}

Here are some handwritten notes about direct sums in a bit more detail:
{|
|[[File:MAT1100 Lec_17, 14-11-10M - direct sums, small p1.pdf]]
|[[File:MAT1100 Lec 17, 14-11-10M - direct sums, small p2.pdf]]
|}
Here are some handwritten notes about presentation matrices for finitely generated modules in a bit more detail:

Apologies! I found a few mistakes in these: there are a few places where I've flipped <math>R^n</math> with X, and other similar errors. I'll post a corrected set soon, although there may still be other errors...
{|
|[[File:14-1100 Lec 17, 14-11-10M - module generated by a matrix, small p1.pdf]]
|[[File:14-1100 Lec 17, 14-11-10M - module generated by a matrix, small p2.pdf]]
|}

{|
|[[File:14-1100 Lec 17 board 1 flip.JPG|200px|thumb|left|Lec 17 board 1]]
|[[File:14-1100 Lec 17 board 2 flip.JPG|200px|thumb|left|Lec 17 board 2]]
|}
{|
|[[File:14-1100 Lec 17 board 3 flip.JPG|200px|thumb|left|Lec 17 board 3]]
|[[File:14-1100 Lec 17 board 4 flip.JPG|200px|thumb|left|Lec 17 board 4]]
|}
{|
|[[File:14-1100 Lec 17 board 5 flip.JPG|200px|thumb|left|Lec 17 board 5]]
|[[File:14-1100 Lec 17 board 6 flip.JPG|200px|thumb|left|Lec 17 board 6]]
|}
{|
|[[File:14-1100 Lec 17 board 7 flip.JPG|200px|thumb|left|Lec 17 board 7]]
|[[File:14-1100 Lec 17 board 8 flip.JPG|200px|thumb|left|Lec 17 board 8]]
|}
{|
|[[File:14-1100 Lec 17 board 9 flip.JPG|200px|thumb|left|Lec 17 board 9]]
|[[File:14-1100 Lec 17 board 10 flip.JPG|200px|thumb|left|Lec 17 board 10]]
|}
{|
|[[File:14-1100 Lec 17 board 11 flip.JPG|200px|thumb|left|Lec 17 board 11]]
|[[File:14-1100 Lec 17 board 12 flip.JPG|200px|thumb|left|Lec 17 board 12]]
|}
{|
|[[File:14-1100 Lec 17 board 13 flip.JPG|200px|thumb|left|Lec 17 board 13]]
|[[File:14-1100 Lec 17 board 14 flip.JPG|200px|thumb|left|Lec 17 board 14]]
|}
{|
|[[File:14-1100 Lec 17 board 15 flip.JPG|200px|thumb|left|Lec 17 board 15]]
|[[File:14-1100 Lec 17 board 16 flip.JPG|200px|thumb|left|Lec 17 board 16]]
|}
{|
|[[File:14-1100 Lec 17 board 17 flip.JPG|200px|thumb|left|Lec 17 board 17]]
|}

'''Here are some handwritten notes:'''

{|
|[[File:14-1100 Lec 17 page 1.JPG|200px|thumb|left|Lec 15 page 1]]
|[[File:14-1100 Lec 17 page 2.JPG|200px|thumb|left|Lec 15 page 2]]
|[[File:14-1100 Lec 17 page 3.JPG|200px|thumb|left|Lec 15 page 3]]
|[[File:14-1100 Lec 17 page 4.JPG|200px|thumb|left|Lec 15 page 4]]
|}
{|
|[[File:14-1100 Lec 17 page 5.JPG|200px|thumb|left|Lec 15 page 5]]
|}

14-1100/Navigation

2014-12-04T16:46:01Z

Anai:

<noinclude>Back to [[14-1100]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center
|colspan=3 style="color: red;"|'''Welcome to Math 1100!'''
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-1100/About This Class|About This Class]]; [[11-1100/Classnotes for Monday September 8|Monday]] - Non Commutative Gaussian Elimination; [[14-1100/Classnotes for Thursday September 11|Thursday]] - the category of groups, automorphisms and conjugations, images and kernels.
|- align=left
|align=center|2
|Sep 15
|[[14-1100/Classnotes for Monday September 15|Monday]] - coset spaces, isomorphism theorems; [[14-1100/Classnotes for Thursday September 18|Thursday]] - simple groups, Jordan-Holder decomposition series.
|- align=left
|align=center|3
|Sep 22
|[[14-1100/Classnotes for Monday September 22|Monday]] - alternating groups, group actions, {{Pensieve Link|Classes/14-1100/The_Simplicity_of_the_Alternating_Groups.html|The Simplicity of the Alternating Groups}}, [[14-1100/Homework Assignment 1| HW1]], [[HW 1 Solutions]], [[14-1100/Class Photo|Class Photo]]; [[14-1100/Classnotes for Thursday September 25|Thursday]] - group actions, Orbit-Stabilizer Thm, Class Equation.
|- align=left
|align=center|4
|Sep 29
|[[14-1100/Classnotes for Monday September 29|Monday]] - Cauchy's Thm, Sylow 1; [[14-1100/Classnotes for Thursday October 2|Thursday]] - Sylow 2.
|- align=left
|align=center|5
|Oct 6
|[[14-1100/Classnotes for Monday October 6|Monday]] - Sylow 3, semi-direct products, braids; [[14-1100/Homework Assignment 2|HW2]]; [[HW 2 Solutions]]; [[14-1100/Classnotes for Thursday October 9|Thursday]] - braids, groups of order 12, [http://matematita.science.unitn.it/braids/index.html Braids]
|- align=left
|align=center|6
|Oct 13
|No class Monday (Thanksgiving); [[14-1100/Classnotes for Thursday October 16|Thursday]] - groups of order 12 cont'd.
|- align=left
|align=center|7
|Oct 20
|[[14-1100/Term Test|Term Test]] on Monday, [[14-1100/Homework Assignment 3|HW3]]; [[HW 3 Solutions]]; [[14-1100/Classnotes for Thursday October 23|Thursday]] - solvable groups, rings: defn's & examples.
|- align=left
|align=center|8
|Oct 27
|[[14-1100/Classnotes for Monday October 27|Monday]] - functors, Cayley-Hamilton Thm, ideals, iso thm 1; [[14-1100/Classnotes for Thursday October 30|Thursday]] - iso thms 2-4, integral domains, maximal ideals, {{Pensieve Link|Classes/14-1100/1t3c5w.pdf|One Theorem, Three Corollaries, Five Weeks}}
|- align=left
|align=center|9
|Nov 3
|[[14-1100/Classnotes for Monday November 3|Monday]] - prime ideals, primes & irreducibles, UFD's, Euc.Domain<math>\Rightarrow</math>PID, [[14-1100/Classnotes for Thursday November 6|Thursday]] - Noetherian rings, PID<math>\Rightarrow</math>UFD, Euclidean Algorithm, modules: defn & examples, [[14-1100/Homework Assignment 4|HW4]], [[HW 4 Solutions]]
|- align=left
|align=center|10
|Nov 10
|[[14-1100/Classnotes for Monday November 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, [[14-1100/Classnotes for Thursday November 13|Thursday]] - row & column reductions plus, existence part of Thm 1 in 1t3c5w handout.
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT's Fall Break, [[14-1100/Homework Assignment 5|HW5]], [[14-1100/Classnotes for Thursday November 20|Thursday]] - 1t3c5w handout cont'd, JCF Tricks & Programs handout
|- align=left
|align=center|12
|Nov 24
|[[14-1100/Classnotes for Monday November 24|Monday]] - JCF Tricks & Programs cont'd, tensor products, [[14-1100/Classnotes for Thursday November 27|Thursday]] - tensor products cont'd
|- align=left
|align=center|13
|Dec 1
|[[14-1100/End-of-Course Schedule|End-of-Course Schedule]]; [[14-1100/Classnotes for Monday December 1|Monday]] - tensor products finale, extension/reduction of scalars, uniqueness part of Thm 1 in 1t3c5w, localization & fields of fractions; [[14-1100/Classnotes for Wednesday December 3|Wednesday]] is a "makeup Monday"!
|- align=left
|colspan=3 align=center|[[14-1100/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-1100-ClassPhoto.jpg|310px|Class Photo]] [[14-1100/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-1100-Splash.png|310px]] See {{Home Link|Talks/Cambridge-1301/|Non Commutative Gaussian Elimination}}
|}

14-1100/Classnotes for Wednesday December 3

2014-12-04T05:36:55Z

Anai:

{{14-1100/Navigation}}

The first half of today's class followed a similar class I gave last year in another course. See the old class {{Dbnvp link|AKT-140314.php|on video!}}

Today's handout is {{Pensieve link|Classes/14-1100/nb/UnifiedJCF.pdf|UnifiedJCF.pdf}}.

{{14-1100:Dror/Students Divider}}

{|
|[[File:14-1100 Lec 23 board 1 flip.JPG|200px|thumb|left|Lec 23 board 1]]
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{|
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|}
{|
|[[File:14-1100 Lec 23 board 7 flip.JPG|200px|thumb|left|Lec 23 board 7]]

'''Here are some handwritten notes:'''

File:14-1100 Lec 23 board 7 flip.JPG

2014-12-04T05:35:35Z

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File:14-1100 Lec 23 board 6 flip.JPG

2014-12-04T05:35:23Z

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File:14-1100 Lec 23 board 5 flip.JPG

2014-12-04T05:35:12Z

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File:14-1100 Lec 23 board 4 flip.JPG

2014-12-04T05:35:02Z

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File:14-1100 Lec 23 board 3 flip.JPG

2014-12-04T05:34:49Z

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File:14-1100 Lec 23 board 2 flip.JPG

2014-12-04T05:34:36Z

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File:14-1100 Lec 23 board 1 flip.JPG

2014-12-04T05:34:24Z

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14-1100/Homework Assignment 5

2014-12-01T20:53:06Z

Anai:

{{14-1100/Navigation}}

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>M</math> be a module over a PID <math>R</math>. Assume that <math>M</math> is isomorphic to <math>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>a_i</math> non-zero non-units and with <math>a_1\mid a_2\mid\cdots\mid a_l</math>. Assume also that <math>M</math> is isomorphic to <math>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>b_i</math> non-zero non-units and with <math>b_1\mid b_2\mid\cdots\mid b_l</math>. Prove that <math>k=m</math>, that <math>l=n</math>, and that <math>a_i\sim b_i</math> for each <math>i</math>.

'''Problem 2.''' Let <math>q</math> and <math>p</math> be primes in a PID <math>R</math> such that <math>p\not\sim q</math>, let <math>\hat{p}</math> denote the operation of "multiplication by <math>p</math>", acting on any <math>R</math>-module <math>M</math>, and let <math>s</math> and <math>t</math> be positive integers.
# For each of the <math>R</math>-modules <math>R</math>, <math>R/\langle q^t\rangle</math>, and <math>R/\langle p^t\rangle</math>, determine <math>\ker\hat{p}^s</math> and <math>(R/\langle p\rangle)\otimes\ker\hat{p}^s</math>.
# 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>{\mathbb C}[t]</math> modules <math>{\mathbb C}[t,t^{-1}]</math> ("Laurent polynomials in <math>t</math>") and <math>{\mathbb C}</math> (here <math>t</math> acts on <math>{\mathbb C}</math> as <math>0</math>).

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

'''Definition.''' The "rank" of a module <math>M</math> over a (commutative) domain <math>R</math> is the maximal number of <math>R</math>-linearly-independent elements of <math>M</math>. (Linear dependence and independence is defined as in vector spaces).

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

'''Problem 5.''' (Dummit and Foote, page 468) Let <math>M</math> be a module over a commutative domain <math>R</math>.
# Suppose that <math>M</math> has rank <math>n</math> and that <math>x_1,\ldots x_n</math> is a maximal set of linearly independent elements of <math>M</math>. Show that <math>\langle x_1,\ldots x_n\rangle</math> is isomorphic to <math>R^n</math> and that <math>M/\langle x_1,\ldots x_n\rangle</math> is a torsion module.
# Conversely show that if <math>M</math> contains a submodule <math>N</math> which is isomorphic to <math>R^n</math> for some <math>n</math>, and so that <math>M/N</math> is torsion, then the rank of <math>M</math> is <math>n</math>.

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

14-1100/Navigation

2014-12-01T20:52:10Z

Anai:

<noinclude>Back to [[14-1100]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center
|colspan=3 style="color: red;"|'''Welcome to Math 1100!'''
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-1100/About This Class|About This Class]]; [[11-1100/Classnotes for Monday September 8|Monday]] - Non Commutative Gaussian Elimination; [[14-1100/Classnotes for Thursday September 11|Thursday]] - the category of groups, automorphisms and conjugations, images and kernels.
|- align=left
|align=center|2
|Sep 15
|[[14-1100/Classnotes for Monday September 15|Monday]] - coset spaces, isomorphism theorems; [[14-1100/Classnotes for Thursday September 18|Thursday]] - simple groups, Jordan-Holder decomposition series.
|- align=left
|align=center|3
|Sep 22
|[[14-1100/Classnotes for Monday September 22|Monday]] - alternating groups, group actions, {{Pensieve Link|Classes/14-1100/The_Simplicity_of_the_Alternating_Groups.html|The Simplicity of the Alternating Groups}}, [[14-1100/Homework Assignment 1| HW1]], [[HW 1 Solutions]], [[14-1100/Class Photo|Class Photo]]; [[14-1100/Classnotes for Thursday September 25|Thursday]] - group actions, Orbit-Stabilizer Thm, Class Equation.
|- align=left
|align=center|4
|Sep 29
|[[14-1100/Classnotes for Monday September 29|Monday]] - Cauchy's Thm, Sylow 1; [[14-1100/Classnotes for Thursday October 2|Thursday]] - Sylow 2.
|- align=left
|align=center|5
|Oct 6
|[[14-1100/Classnotes for Monday October 6|Monday]] - Sylow 3, semi-direct products, braids; [[14-1100/Homework Assignment 2|HW2]]; [[HW 2 Solutions]]; [[14-1100/Classnotes for Thursday October 9|Thursday]] - braids, groups of order 12, [http://matematita.science.unitn.it/braids/index.html Braids]
|- align=left
|align=center|6
|Oct 13
|No class Monday (Thanksgiving); [[14-1100/Classnotes for Thursday October 16|Thursday]] - groups of order 12 cont'd.
|- align=left
|align=center|7
|Oct 20
|[[14-1100/Term Test|Term Test]] on Monday, [[14-1100/Homework Assignment 3|HW3]]; [[HW 3 Solutions]]; [[14-1100/Classnotes for Thursday October 23|Thursday]] - solvable groups, rings: defn's & examples.
|- align=left
|align=center|8
|Oct 27
|[[14-1100/Classnotes for Monday October 27|Monday]] - functors, Cayley-Hamilton Thm, ideals, iso thm 1; [[14-1100/Classnotes for Thursday October 30|Thursday]] - iso thms 2-4, integral domains, maximal ideals, {{Pensieve Link|Classes/14-1100/1t3c5w.pdf|One Theorem, Three Corollaries, Five Weeks}}
|- align=left
|align=center|9
|Nov 3
|[[14-1100/Classnotes for Monday November 3|Monday]] - prime ideals, primes & irreducibles, UFD's, Euc.Domain<math>\Rightarrow</math>PID, [[14-1100/Classnotes for Thursday November 6|Thursday]] - Noetherian rings, PID<math>\Rightarrow</math>UFD, Euclidean Algorithm, modules: defn & examples, [[14-1100/Homework Assignment 4|HW4]], [[HW 4 Solutions]]
|- align=left
|align=center|10
|Nov 10
|[[14-1100/Classnotes for Monday November 10|Monday]] - R is a PID iff R has a D-H norm, R-modules, direct sums, every <math>n \times X</math> matrix A defines a f.g. module, [[14-1100/Classnotes for Thursday November 13|Thursday]] - row & column reductions plus, existence part of Thm 1 in 1t3c5w handout.
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT's Fall Break, [[14-1100/Homework Assignment 5|HW5]], [[14-1100/Classnotes for Thursday November 20|Thursday]] - 1t3c5w handout cont'd, JCF Tricks & Programs handout
|- align=left
|align=center|12
|Nov 24
|[[14-1100/Classnotes for Monday November 24|Monday]] - JCF Tricks & Programs cont'd, tensor products, [[14-1100/Classnotes for Thursday November 27|Thursday]] - tensor products cont'd
|- align=left
|align=center|13
|Dec 1
|[[14-1100/End-of-Course Schedule|End-of-Course Schedule]]; Wednesday is a "makeup Monday"! [[14-1100/Classnotes for Monday December 1|Monday]] - tensor products finale, extension/reduction of scalars, uniqueness part of Thm 1 in 1t3c5w, localization & fields of fractions
|- align=left
|colspan=3 align=center|[[14-1100/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-1100-ClassPhoto.jpg|310px|Class Photo]] [[14-1100/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-1100-Splash.png|310px]] See {{Home Link|Talks/Cambridge-1301/|Non Commutative Gaussian Elimination}}
|}

14-1100/Navigation

2014-12-01T20:48:38Z

Anai:

<noinclude>Back to [[14-1100]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center
|colspan=3 style="color: red;"|'''Welcome to Math 1100!'''
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-1100/About This Class|About This Class]]; [[11-1100/Classnotes for Monday September 8|Monday]] - Non Commutative Gaussian Elimination; [[14-1100/Classnotes for Thursday September 11|Thursday]] - the category of groups, automorphisms and conjugations, images and kernels.
|- align=left
|align=center|2
|Sep 15
|[[14-1100/Classnotes for Monday September 15|Monday]] - coset spaces, isomorphism theorems; [[14-1100/Classnotes for Thursday September 18|Thursday]] - simple groups, Jordan-Holder decomposition series.
|- align=left
|align=center|3
|Sep 22
|[[14-1100/Classnotes for Monday September 22|Monday]] - alternating groups, group actions, {{Pensieve Link|Classes/14-1100/The_Simplicity_of_the_Alternating_Groups.html|The Simplicity of the Alternating Groups}}, [[14-1100/Homework Assignment 1| HW1]], [[HW 1 Solutions]], [[14-1100/Class Photo|Class Photo]]; [[14-1100/Classnotes for Thursday September 25|Thursday]] - group actions, Orbit-Stabilizer Thm, Class Equation.
|- align=left
|align=center|4
|Sep 29
|[[14-1100/Classnotes for Monday September 29|Monday]] - Cauchy's Thm, Sylow 1; [[14-1100/Classnotes for Thursday October 2|Thursday]] - Sylow 2.
|- align=left
|align=center|5
|Oct 6
|[[14-1100/Classnotes for Monday October 6|Monday]] - Sylow 3, semi-direct products, braids; [[14-1100/Homework Assignment 2|HW2]]; [[HW 2 Solutions]]; [[14-1100/Classnotes for Thursday October 9|Thursday]] - braids, groups of order 12, [http://matematita.science.unitn.it/braids/index.html Braids]
|- align=left
|align=center|6
|Oct 13
|No class Monday (Thanksgiving); [[14-1100/Classnotes for Thursday October 16|Thursday]] - groups of order 12 cont'd.
|- align=left
|align=center|7
|Oct 20
|[[14-1100/Term Test|Term Test]] on Monday, [[14-1100/Homework Assignment 3|HW3]]; [[HW 3 Solutions]]; [[14-1100/Classnotes for Thursday October 23|Thursday]] - solvable groups, rings: defn's & examples.
|- align=left
|align=center|8
|Oct 27
|[[14-1100/Classnotes for Monday October 27|Monday]] - functors, Cayley-Hamilton Thm, ideals, iso thm 1; [[14-1100/Classnotes for Thursday October 30|Thursday]] - iso thms 2-4, integral domains, maximal ideals, {{Pensieve Link|Classes/14-1100/1t3c5w.pdf|One Theorem, Three Corollaries, Five Weeks}}
|- align=left
|align=center|9
|Nov 3
|[[14-1100/Classnotes for Monday November 3|Monday]] - prime ideals, primes & irreducibles, UFD's, Euc.Domain<math>\Rightarrow</math>PID, [[14-1100/Classnotes for Thursday November 6|Thursday]] - Noetherian rings, PID<math>\Rightarrow</math>UFD, Euclidean Algorithm, modules: defn & examples, [[14-1100/Homework Assignment 4|HW4]], [[HW 4 Solutions]]
|- align=left
|align=center|10
|Nov 10
|[[14-1100/Classnotes for Monday November 10|Monday]] - R is a PID iff R has a D-H norm, R-modules, direct sums, every <math>n \times X</math> matrix A defines a f.g. module, [[14-1100/Classnotes for Thursday November 13|Thursday]] - row & column reductions plus, existence part of Thm 1 in 1t3c5w handout.
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT's Fall Break, [[14-1100/Homework Assignment 5|HW5]], [[14-1100/Classnotes for Thursday November 20|Thursday]] - 1t3c5w handout cont'd, JCF Tricks & Programs handout
|- align=left
|align=center|12
|Nov 24
|[[14-1100/Classnotes for Monday November 24|Monday]] - JCF Tricks & Programs cont'd, tensor products, [[14-1100/Classnotes for Thursday November 27|Thursday]] - tensor products cont'd
|- align=left
|align=center|13
|Dec 1
|[[14-1100/End-of-Course Schedule|End-of-Course Schedule]]; Wednesday is a "makeup Monday"! [[14-1100/Classnotes for Monday December 1|Monday]]
|- align=left
|colspan=3 align=center|[[14-1100/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-1100-ClassPhoto.jpg|310px|Class Photo]] [[14-1100/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-1100-Splash.png|310px]] See {{Home Link|Talks/Cambridge-1301/|Non Commutative Gaussian Elimination}}
|}

14-1100/Classnotes for Monday December 1

2014-12-01T20:46:22Z

Anai:

{{14-1100/Navigation}}

Note: HW5 has been extended and is due in Dror's mailbox by the end of Monday, December 8.
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14-1100/Classnotes for Monday December 1

2014-12-01T20:45:29Z

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14-1100/End-of-Course Schedule

2014-12-01T19:22:58Z

Anai:

{{14-1100/Navigation}}

Here's a tentative activity schedule for the last few days of our class:

* Wednesday December 3, 1-3PM: Our last class. Note the unusual time - that Wednesday is a "UofT Monday".

* Wednesday December 3, 3PM-4PM: Office hour.

* Thursday December 4, 10AM-12PM: Riddles session with {{Dror}} at Bahen 6183. Not a part of this class! Not mandatory in any way! Also with the students of [[14-240]].

* Monday December 8: [[14-1100/Homework Assignment 5|HW5]] is due! Hand it in to Dror's mailbox by the end of the day/by the beginning of Tuesday. (Extended from Dec 3rd)

* Sunday December 14, noon: At 24 hours before the final, edits to this class' web site no longer count for good deed points.

* Sunday December 14, 2-5PM: Pre-final office hours.

* Monday December 15, 10-11AM: Pre-final office hour.

* Monday December 15, 12-3PM: [[14-1100/The Final Exam|Our final exam]] at Bahen 6183.

'''Warning.''' Last minute changes are possible! Check here before you go.

14-1100/Homework Assignment 5

2014-12-01T19:11:25Z

Anai:

{{14-1100/Navigation}}

This assignment is extended from class time on Wednesday, December 3, 2014 (a "virtual Monday" and the last day of the semester) to Monday, December 8 in Dror's mailbox.

===Solve the following questions===

'''Problem 1.''' Let <math>M</math> be a module over a PID <math>R</math>. Assume that <math>M</math> is isomorphic to <math>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>a_i</math> non-zero non-units and with <math>a_1\mid a_2\mid\cdots\mid a_l</math>. Assume also that <math>M</math> is isomorphic to <math>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>b_i</math> non-zero non-units and with <math>b_1\mid b_2\mid\cdots\mid b_l</math>. Prove that <math>k=m</math>, that <math>l=n</math>, and that <math>a_i\sim b_i</math> for each <math>i</math>.

'''Problem 2.''' Let <math>q</math> and <math>p</math> be primes in a PID <math>R</math> such that <math>p\not\sim q</math>, let <math>\hat{p}</math> denote the operation of "multiplication by <math>p</math>", acting on any <math>R</math>-module <math>M</math>, and let <math>s</math> and <math>t</math> be positive integers.
# For each of the <math>R</math>-modules <math>R</math>, <math>R/\langle q^t\rangle</math>, and <math>R/\langle p^t\rangle</math>, determine <math>\ker\hat{p}^s</math> and <math>(R/\langle p\rangle)\otimes\ker\hat{p}^s</math>.
# 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>{\mathbb C}[t]</math> modules <math>{\mathbb C}[t,t^{-1}]</math> ("Laurent polynomials in <math>t</math>") and <math>{\mathbb C}</math> (here <math>t</math> acts on <math>{\mathbb C}</math> as <math>0</math>).

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

'''Definition.''' The "rank" of a module <math>M</math> over a (commutative) domain <math>R</math> is the maximal number of <math>R</math>-linearly-independent elements of <math>M</math>. (Linear dependence and independence is defined as in vector spaces).

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

'''Problem 5.''' (Dummit and Foote, page 468) Let <math>M</math> be a module over a commutative domain <math>R</math>.
# Suppose that <math>M</math> has rank <math>n</math> and that <math>x_1,\ldots x_n</math> is a maximal set of linearly independent elements of <math>M</math>. Show that <math>\langle x_1,\ldots x_n\rangle</math> is isomorphic to <math>R^n</math> and that <math>M/\langle x_1,\ldots x_n\rangle</math> is a torsion module.
# Conversely show that if <math>M</math> contains a submodule <math>N</math> which is isomorphic to <math>R^n</math> for some <math>n</math>, and so that <math>M/N</math> is torsion, then the rank of <math>M</math> is <math>n</math>.

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

14-1100/Navigation

2014-11-28T18:56:09Z

Anai:

<noinclude>Back to [[14-1100]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center
|colspan=3 style="color: red;"|'''Welcome to Math 1100!'''
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-1100/About This Class|About This Class]]; [[11-1100/Classnotes for Monday September 8|Monday]] - Non Commutative Gaussian Elimination; [[14-1100/Classnotes for Thursday September 11|Thursday]] - the category of groups, automorphisms and conjugations, images and kernels.
|- align=left
|align=center|2
|Sep 15
|[[14-1100/Classnotes for Monday September 15|Monday]] - coset spaces, isomorphism theorems; [[14-1100/Classnotes for Thursday September 18|Thursday]] - simple groups, Jordan-Holder decomposition series.
|- align=left
|align=center|3
|Sep 22
|[[14-1100/Classnotes for Monday September 22|Monday]] - alternating groups, group actions, {{Pensieve Link|Classes/14-1100/The_Simplicity_of_the_Alternating_Groups.html|The Simplicity of the Alternating Groups}}, [[14-1100/Homework Assignment 1| HW1]], [[HW 1 Solutions]], [[14-1100/Class Photo|Class Photo]]; [[14-1100/Classnotes for Thursday September 25|Thursday]] - group actions, Orbit-Stabilizer Thm, Class Equation.
|- align=left
|align=center|4
|Sep 29
|[[14-1100/Classnotes for Monday September 29|Monday]] - Cauchy's Thm, Sylow 1; [[14-1100/Classnotes for Thursday October 2|Thursday]] - Sylow 2.
|- align=left
|align=center|5
|Oct 6
|[[14-1100/Classnotes for Monday October 6|Monday]] - Sylow 3, semi-direct products, braids; [[14-1100/Homework Assignment 2|HW2]]; [[HW 2 Solutions]]; [[14-1100/Classnotes for Thursday October 9|Thursday]] - braids, groups of order 12, [http://matematita.science.unitn.it/braids/index.html Braids]
|- align=left
|align=center|6
|Oct 13
|No class Monday (Thanksgiving); [[14-1100/Classnotes for Thursday October 16|Thursday]] - groups of order 12 cont'd.
|- align=left
|align=center|7
|Oct 20
|[[14-1100/Term Test|Term Test]] on Monday, [[14-1100/Homework Assignment 3|HW3]]; [[HW 3 Solutions]]; [[14-1100/Classnotes for Thursday October 23|Thursday]] - solvable groups, rings: defn's & examples.
|- align=left
|align=center|8
|Oct 27
|[[14-1100/Classnotes for Monday October 27|Monday]] - functors, Cayley-Hamilton Thm, ideals, iso thm 1; [[14-1100/Classnotes for Thursday October 30|Thursday]] - iso thms 2-4, integral domains, maximal ideals, {{Pensieve Link|Classes/14-1100/1t3c5w.pdf|One Theorem, Three Corollaries, Five Weeks}}
|- align=left
|align=center|9
|Nov 3
|[[14-1100/Classnotes for Monday November 3|Monday]] - prime ideals, primes & irreducibles, UFD's, Euc.Domain<math>\Rightarrow</math>PID, [[14-1100/Classnotes for Thursday November 6|Thursday]] - Noetherian rings, PID<math>\Rightarrow</math>UFD, Euclidean Algorithm, modules: defn & examples, [[14-1100/Homework Assignment 4|HW4]], [[HW 4 Solutions]]
|- align=left
|align=center|10
|Nov 10
|[[14-1100/Classnotes for Monday November 10|Monday]] - R is a PID iff R has a D-H norm, R-modules, direct sums, every <math>n \times X</math> matrix A defines a f.g. module, [[14-1100/Classnotes for Thursday November 13|Thursday]] - row & column reductions plus, existence part of Thm 1 in 1t3c5w handout.
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT's Fall Break, [[14-1100/Homework Assignment 5|HW5]], [[14-1100/Classnotes for Thursday November 20|Thursday]] - 1t3c5w handout cont'd, JCF Tricks & Programs handout
|- align=left
|align=center|12
|Nov 24
|[[14-1100/Classnotes for Monday November 24|Monday]] - JCF Tricks & Programs cont'd, tensor products, [[14-1100/Classnotes for Thursday November 27|Thursday]] - tensor products cont'd
|- align=left
|align=center|13
|Dec 1
|Wednesday is a "makeup Monday"!
|- align=left
|colspan=3 align=center|[[14-1100/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-1100-ClassPhoto.jpg|310px|Class Photo]] [[14-1100/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-1100-Splash.png|310px]] See {{Home Link|Talks/Cambridge-1301/|Non Commutative Gaussian Elimination}}
|}

14-1100/Navigation

2014-11-28T18:36:58Z

Anai:

<noinclude>Back to [[14-1100]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center
|colspan=3 style="color: red;"|'''Welcome to Math 1100!'''
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-1100/About This Class|About This Class]]; [[11-1100/Classnotes for Monday September 8|Monday]] - Non Commutative Gaussian Elimination; [[14-1100/Classnotes for Thursday September 11|Thursday]] - the category of groups, automorphisms and conjugations, images and kernels.
|- align=left
|align=center|2
|Sep 15
|[[14-1100/Classnotes for Monday September 15|Monday]] - coset spaces, isomorphism theorems; [[14-1100/Classnotes for Thursday September 18|Thursday]] - simple groups, Jordan-Holder decomposition series.
|- align=left
|align=center|3
|Sep 22
|[[14-1100/Classnotes for Monday September 22|Monday]] - alternating groups, group actions, {{Pensieve Link|Classes/14-1100/The_Simplicity_of_the_Alternating_Groups.html|The Simplicity of the Alternating Groups}}, [[14-1100/Homework Assignment 1| HW1]], [[HW 1 Solutions]], [[14-1100/Class Photo|Class Photo]]; [[14-1100/Classnotes for Thursday September 25|Thursday]] - group actions, Orbit-Stabilizer Thm, Class Equation.
|- align=left
|align=center|4
|Sep 29
|[[14-1100/Classnotes for Monday September 29|Monday]] - Cauchy's Thm, Sylow 1; [[14-1100/Classnotes for Thursday October 2|Thursday]] - Sylow 2.
|- align=left
|align=center|5
|Oct 6
|[[14-1100/Classnotes for Monday October 6|Monday]] - Sylow 3, semi-direct products, braids; [[14-1100/Homework Assignment 2|HW2]]; [[HW 2 Solutions]]; [[14-1100/Classnotes for Thursday October 9|Thursday]] - braids, groups of order 12, [http://matematita.science.unitn.it/braids/index.html Braids]
|- align=left
|align=center|6
|Oct 13
|No class Monday (Thanksgiving); [[14-1100/Classnotes for Thursday October 16|Thursday]] - groups of order 12 cont'd.
|- align=left
|align=center|7
|Oct 20
|[[14-1100/Term Test|Term Test]] on Monday, [[14-1100/Homework Assignment 3|HW3]]; [[HW 3 Solutions]]; [[14-1100/Classnotes for Thursday October 23|Thursday]] - solvable groups, rings: defn's & examples.
|- align=left
|align=center|8
|Oct 27
|[[14-1100/Classnotes for Monday October 27|Monday]] - functors, Cayley-Hamilton Thm, ideals, iso thm 1; [[14-1100/Classnotes for Thursday October 30|Thursday]] - iso thms 2-4, integral domains, maximal ideals, {{Pensieve Link|Classes/14-1100/1t3c5w.pdf|One Theorem, Three Corollaries, Five Weeks}}
|- align=left
|align=center|9
|Nov 3
|[[14-1100/Classnotes for Monday November 3|Monday]] - prime ideals, primes & irreducibles, UFD's, Euc.Domain<math>\Rightarrow</math>PID, [[14-1100/Classnotes for Thursday November 6|Thursday]] - Noetherian rings, PID<math>\Rightarrow</math>UFD, Euclidean Algorithm, modules: defn & examples, [[14-1100/Homework Assignment 4|HW4]], [[HW 4 Solutions]]
|- align=left
|align=center|10
|Nov 10
|[[14-1100/Classnotes for Monday November 10|Monday]] - R is a PID iff R has a D-H norm, R-modules, direct sums, every <math>n \times X</math> matrix A defines a f.g. module, [[14-1100/Classnotes for Thursday November 13|Thursday]]
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT's Fall Break, [[14-1100/Homework Assignment 5|HW5]], [[14-1100/Classnotes for Thursday November 20|Thursday]]
|- align=left
|align=center|12
|Nov 24
|[[14-1100/Classnotes for Monday November 24|Monday]], [[14-1100/Classnotes for Thursday November 27|Thursday]]
|- align=left
|align=center|13
|Dec 1
|Wednesday is a "makeup Monday"!
|- align=left
|colspan=3 align=center|[[14-1100/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-1100-ClassPhoto.jpg|310px|Class Photo]] [[14-1100/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-1100-Splash.png|310px]] See {{Home Link|Talks/Cambridge-1301/|Non Commutative Gaussian Elimination}}
|}

14-1100/Classnotes for Thursday November 20

2014-11-28T18:07:08Z

Anai:

{{14-1100/Navigation}}

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'''Here are some handwritten notes:'''

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2014-11-28T18:06:27Z

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14-1100/Navigation

2014-11-28T18:03:44Z

Anai:

<noinclude>Back to [[14-1100]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center
|colspan=3 style="color: red;"|'''Welcome to Math 1100!'''
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-1100/About This Class|About This Class]]; [[11-1100/Classnotes for Monday September 8|Monday]] - Non Commutative Gaussian Elimination; [[14-1100/Classnotes for Thursday September 11|Thursday]] - the category of groups, automorphisms and conjugations, images and kernels.
|- align=left
|align=center|2
|Sep 15
|[[14-1100/Classnotes for Monday September 15|Monday]] - coset spaces, isomorphism theorems; [[14-1100/Classnotes for Thursday September 18|Thursday]] - simple groups, Jordan-Holder decomposition series.
|- align=left
|align=center|3
|Sep 22
|[[14-1100/Classnotes for Monday September 22|Monday]] - alternating groups, group actions, {{Pensieve Link|Classes/14-1100/The_Simplicity_of_the_Alternating_Groups.html|The Simplicity of the Alternating Groups}}, [[14-1100/Homework Assignment 1| HW1]], [[HW 1 Solutions]], [[14-1100/Class Photo|Class Photo]]; [[14-1100/Classnotes for Thursday September 25|Thursday]] - group actions, Orbit-Stabilizer Thm, Class Equation.
|- align=left
|align=center|4
|Sep 29
|[[14-1100/Classnotes for Monday September 29|Monday]] - Cauchy's Thm, Sylow 1; [[14-1100/Classnotes for Thursday October 2|Thursday]] - Sylow 2.
|- align=left
|align=center|5
|Oct 6
|[[14-1100/Classnotes for Monday October 6|Monday]] - Sylow 3, semi-direct products, braids; [[14-1100/Homework Assignment 2|HW2]]; [[HW 2 Solutions]]; [[14-1100/Classnotes for Thursday October 9|Thursday]] - braids, groups of order 12, [http://matematita.science.unitn.it/braids/index.html Braids]
|- align=left
|align=center|6
|Oct 13
|No class Monday (Thanksgiving); [[14-1100/Classnotes for Thursday October 16|Thursday]] - groups of order 12 cont'd.
|- align=left
|align=center|7
|Oct 20
|[[14-1100/Term Test|Term Test]] on Monday, [[14-1100/Homework Assignment 3|HW3]]; [[HW 3 Solutions]]; [[14-1100/Classnotes for Thursday October 23|Thursday]] - solvable groups, rings: defn's & examples.
|- align=left
|align=center|8
|Oct 27
|[[14-1100/Classnotes for Monday October 27|Monday]] - functors, Cayley-Hamilton Thm, ideals, iso thm 1; [[14-1100/Classnotes for Thursday October 30|Thursday]] - iso thms 2-4, integral domains, maximal ideals, {{Pensieve Link|Classes/14-1100/1t3c5w.pdf|One Theorem, Three Corollaries, Five Weeks}}
|- align=left
|align=center|9
|Nov 3
|[[14-1100/Classnotes for Monday November 3|Monday]] - prime ideals, primes & irreducibles, UFD's, Euc.Domain<math>\Rightarrow</math>PID, [[14-1100/Classnotes for Thursday November 6|Thursday]] - Noetherian rings, PID<math>\Rightarrow</math>UFD, Euclidean Algorithm, modules: defn & examples, [[14-1100/Homework Assignment 4|HW4]], [[HW 4 Solutions]]
|- align=left
|align=center|10
|Nov 10
|[[14-1100/Classnotes for Monday November 10|Monday]], [[14-1100/Classnotes for Thursday November 13|Thursday]]
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT's Fall Break, [[14-1100/Homework Assignment 5|HW5]], [[14-1100/Classnotes for Thursday November 20|Thursday]]
|- align=left
|align=center|12
|Nov 24
|[[14-1100/Classnotes for Monday November 24|Monday]], [[14-1100/Classnotes for Thursday November 27|Thursday]]
|- align=left
|align=center|13
|Dec 1
|Wednesday is a "makeup Monday"!
|- align=left
|colspan=3 align=center|[[14-1100/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-1100-ClassPhoto.jpg|310px|Class Photo]] [[14-1100/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:10-1100-Splash.png|310px]] See {{Home Link|Talks/Cambridge-1301/|Non Commutative Gaussian Elimination}}
|}

14-1100/Classnotes for Thursday November 27

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