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🗺️ Map of Content - Category Theory
🗺️ Map of Content - Modules
10-19 Teaching
11 Classes
MATH 561 - Graduate Algebra
2024 - Fall
Homework
Homework 1
Homework 2
Homework 3
Homework 4
Homework 5
Homework 6
Homework 7
Homework 8
Study Guides
Study Guide for Midterm Exam
Daily class summaries
MATH 561 Home
Midterm Exam Solutions
Exercises
A criterion for a module to be finitely generated
Alternate characterization of rank
Annihilators of torsion modules
Annihilators
Direct sum of free modules
Direct sums and injective, projective, flat
Equalizers and coequalizers of matrices
Equivalence relations on sets
Finite abelian groups are neither injective nor projective
Group morphisms that cannot define module morphisms
Groups as categories
Hom(R,M) is M
Hom(R,R) is R
Irreducible modules
Quotients of cyclic modules
Rank and quotients
Ring property from module property
Submodules via ideals
Tensor product of projective modules is projective
The evaluation map
The pullback functor is an adjoint
The Short Five Lemma
There is no 'center' functor on Grp
Torsion Z-modules
Torsion submodules
20-29 Research
24 Summer REUs
2024
Summer REU 2024
40-49 Knowledge
41 Mathematics
Algebra theory
Algebras
Exterior algebras
Symmetric algebras
Symmetric and alternating tensors
Tensor algebras
Algebraic geometry
Examples of classical conic duality
Calculus
Determining and classifying local extrema
Determining constrained extrema
Category theory
Abelian Categories
Ab-categories
Abelian categories
Additive categories
Chain complexes
Diagram chases without elements
Diagram lemmas
Double complexes and mural maps
Exact sequences
The Salamander Lemma
Adjoints
Adjoints
Examples of adjoints
Properties of adjoints
Basic Structures
Categories
Functor categories
Functors
Natural transformations
Special morphisms
Universal Properties
Universal arrows and elements
Universal Properties I - Inspiring Examples
Universal Properties II - Commutative diagrams, cones and limits
Universal Properties III - Yoneda's Lemma
Field theory
A question about finite fields
Group theory
Normal subgroups
Module theory
Basic definitions and examples
Module morphisms and submodules
Module morphisms
Modules
Submodules
Bimodules
Bimodule morphisms
Bimodules
The 2-category of bimodules
Constructions on modules
Direct products of modules
Direct products vs. direct sums vs. sums
Direct sums of modules
Examples of free modules
Free modules
Generators for modules and submodules
Quotient modules
Sums of submodules
The Isomorphism Theorems for Modules
Exact sequences
Exact Sequences I - Illustrative Examples
Exact Sequences II - Exact Sequences
Exact Sequences III - Morphisms of Exact Sequences
Exact Sequences IV - Exact Sequences and Functors
Modules over a PID
Jordan Canonical Form I - Definition
Jordan Canonical Form II - Computation
Linear independence, rank and the structure of free modules
Modules over a PID - The Fundamental Theorem
Noetherian modules
Rational Canonical Form I - Definition
Rational Canonical Form II - Additional Properties
Rational Canonical Form III - Computation
Tensor products of modules
Tensor Products I - Extending scalars
Tensor Products II - Tensor products of bimodules
Tensor Products III - Balanced Maps and a Universal Property of the Tensor Product
Tensor Products IV - The Adjoint Property
Precalculus
MATH 118 - Section 1.6 Hints
Ring theory
Chinese Remainder Theorem
Graded rings
Least common multiples
Principal ideal domains (PIDs)
Tropical algebraic geometry
Investigation into tropical tangency loci
Linear tropical varieties
Quadratic tropical varieties
Tropical varieties
42 Software
LaTeX
Systems of equations
47 Quotes
Mathematical Quotes
50-59 Logs
51 Class summaries
2024 - Fall
MATH 561
2024-09
2024-09-23
2024-09-24
2024-09-26
2024-09-27
2024-09-30
2024-10
2024-10-01
2024-10-03
2024-10-04
2024-10-07
2024-10-08
2024-10-10
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2024-10-22
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2024-10-28
2024-10-31
2024-29-10
2024-11
2024-11-01
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2024-11-05
2024-11-07
2024-11-08
2024-11-12
2024-11-14
2024-11-15
2024-11-18
2024-11-19
52 Research meetings
2024 - Summer
REU Meeting - 2024-07-02
REU Meeting - 2024-07-05
REU Meeting - 2024-07-09
REU Meeting - 2024-07-12
REU Meeting - 2024-07-16
REU Meeting - 2024-07-19
REU Meeting - 2024-07-23
REU Meeting - 2024-07-26
REU Meeting - 2024-07-30
REU Meeting - 2024-08-02
REU Meeting - 2024-08-06
REU Meeting - 2024-08-09
REU Meeting - 2024-08-13
REU Meeting - 2024-08-20
REU Meeting - 2024-08-23
REU Meeting - 2024-08-27
Home
Direct sum of free modules
Prove that any direct sum of free
R
-modules is free.