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00-09 Atlas
🗺️ Map of Content - Algebra Qual
🗺️ Map of Content - Algebra Theory
🗺️ Map of Content - Category Theory
🗺️ Map of Content - Modules
🗺️ Map of Content - Ring Theory
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 Final Exam
Study Guide for Midterm Exam
Daily class summaries
MATH 561 Home
Midterm Exam Solutions
Exercise Solutions
Annihilators - Solution
Exercises
A criterion for a module to be finitely generated
Alternate characterization of rank
An intramural isomorphism
Annihilators of torsion modules
Annihilators
Cokernels in the category of abelian groups
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
Primes and annihilators
Quotients of cyclic modules
Rank and quotients
Rational and Jordan canonical forms
Ring property from module property
Submodules via ideals
Tensor product of projective modules is projective
Tensor product of rings is a coproduct
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
14 Algebra Qual
Previous exams
Algebra Qual 2014-03
Algebra Qual 2014-09
Algebra Qual 2015-03
Algebra Qual 2015-09
Algebra Qual 2016-03
Algebra Qual 2016-09
Algebra Qual 2017-03
Algebra Qual 2017-09
Algebra Qual 2018-06
Algebra Qual 2018-09
Algebra Qual 2019-01
Algebra Qual 2019-06
Algebra Qual 2019-09
Algebra Qual 2020-01
Algebra Qual 2020-06
Algebra Qual 2021-01
Algebra Qual 2021-06
Algebra Qual 2021-09
Algebra Qual 2022-01
Algebra Qual 2022-06
Algebra Qual 2022-09
Algebra Qual 2023-01
Algebra Qual 2023-06
Algebra Qual 2023-09
Algebra Qual 2024-01
Algebra Qual 2024-06
Algebra Qual 2024-09
Algebra Qual 2025-01
Problem bank
Pool problems
Group theory
A condition to be non-cyclic
A condition under which a group must be abelian (2)
A dihedral group that is not an internal direct product
A group isomorphic to a subgroup of a direct product of quotient groups
A group isomorphic to an internal direct product
A group of upper-triangular matrices
A group with a trivial automorphism group (2)
A group with a trivial automorphism group
A property of the order of an element
An automorphism of a group of odd order
Another condition for a group to be abelian
Automorphisms of a finite cyclic group
Closely related subgroups of a finite group
Comparing cosets
Computations with inner automorphisms
Elements of finite order
Elements of order 2
Existence of a normal subgroup of finite index
Existence of an identity element in a group
Existence of automorphisms
Image of a normal subgroup and induced morphisms
Image of the identity is the identity
Indices and intersections
Inner automorphisms and the center of a group
Nonexistence of a simple group of a given order
Nonexistence of small nonabelian groups
Normal subgroups with trivial intersection
Normality and the operation on cosets (defunct)
Normality and the operation on cosets
Normalizer of a subgroup
Normalizers and centralizers
Order of a power of an element
Order of an element in a finite group
Orders of elements in a quotient group
Preimage of a subgroup
Product of two subgroups
Products of quotient groups
Projection onto a quotient
Properties of the center of a group
Property of the order of an element
Proving Lagrange's Theorem
Stabilizer of a coset
Subgroups of a group of even order
The integers as a subgroup of the rationals
The Third Isomorphism Theorem
Linear algebra
A condition ensuring diagonalizability
A property of surjective linear transformations
An upper bound on the number of nonzero eigenvalues
Analyzing an unusual matrix
Direct sums and idempotent transformations
Dot product and cross product as linear transformations
Eigenvalues and eigenspaces of a matrix with a given property
Eigenvectors and linear independence
Eigenvectors of commuting linear transformations
Eigenvectors with distinct eigenvalues are linearly independent
Linear dependence and relations
Linear endomorphism of a vector space of matrices
Matrix representing a linear transformation
Numerical range of a linear transformation
Projections and adjoints
Properties of transpose
Skew-symmetric matrices
Sum and union of subspaces
Ring theory
A non-PID
A ring in which all prime ideals are maximal
Automorphisms of a ring
Boolean algebras
Boolean rings are commutative
Characteristic of a ring (2)
Characteristic of a ring
Dimension of a PID
Euclidean domains are PIDs
Existence of an identity element in a finite ring
Generator for a field extension
Ideals in a polynomial ring (2)
Ideals in a polynomial ring
Idempotent elements in a ring (2)
Idempotent elements in a ring
Image of the identity element under a ring morphism
Maximal ideals in a PID
Nilpotent elements in a ring
Nilpotent elements
Nonzero prime ideals are maximal in a PID
Polynomials with even constant term
Prime and irreducible elements in a commutative ring
Prime ideals and quotient rings
Properties of Boolean rings
Properties of the annihilator
Proving an ideal is prime
Quotienting out nilpotent elements
Quotients and direct products
The nilradical of a ring
The structure of the integers as both a group and a ring
Pool problems in group theory
Pool problems in linear algebra
Pool problems in ring theory
Template problems
Group theory
A condition under which a group must be abelian
Computations in symmetric groups
Computing an automorphism group
Counting morphisms between specified groups
Directly proving the existence of an element of a desired order
Finding all morphisms between two groups
Inner and outer automorphisms
Inner automorphisms of an alternating group
Nonexistence of morphisms between two groups
Order of elements in a symmetric group
The cyclic group of order 2020
Verifying axioms of a group
Working modulo 11
Linear algebra
A linear transformation from a vector space of polynomials
A vector space of square matrices
Bases for a subspace and its orthogonal complement
Computations with a given linear transformation
Diagonalization and matrix powers
Diagonalization of a given matrix (2)
Diagonalization of a given matrix
Dimension of a subspace and its orthogonal complement
Jordan canonical form of a matrix (2)
Jordan canonical form of a matrix
Linear transformation from a vector space of polynomials
Matrix and characteristic polynomial for a given linear transformation
Matrix and eigenvalues of a given linear transformation
Orthogonal complements
Orthogonal projection and scaling
Orthogonal projection onto a line (2)
Orthogonal projection onto a line (3)
Orthogonal projection onto a line
Orthogonal projection onto a plane (2)
Orthogonal projection onto a plane
Projection onto a plane
Radial expansion from a fixed line
Reflection across a plane
Rotation around an axis
Ring theory
A maximal ideal in a function ring
An evaluation morphism
An isomorphism of rings
Constructing a field extension
Constructing the field with eight elements
Evaluation at i
Existence of certain ring morphisms
Group of units of a product
Image of an evaluation morphism
Morphism from the Gaussian integers (2)
Morphism from the Gaussian integers
The field with eight elements
The field with nine elements
The kernel of an evaluation morphism
Using the Chinese Remainder Theorem
Template problems in group theory
Template problems in linear algebra
Template problems in ring theory
Notation Key
Past exams
Problem bank
Syllabus
The Algebra Qual
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
Abelian categories
Additive categories
Chain complexes
Diagram chases without elements
Diagram lemmas
Double complexes and mural maps
Exact sequences and chain homology
Preadditive categories
Spectral 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
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
2024-10-11
2024-10-14
2024-10-15
2024-10-17
2024-10-18
2024-10-21
2024-10-22
2024-10-24
2024-10-28
2024-10-31
2024-29-10
2024-11
2024-11-01
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2024-11-08
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2024-11-18
2024-11-19
2024-11-21
2024-11-22
2024-12
2024-12-03
2024-12-05
2024-12-06
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
🗺️ Map of Content - Ring Theory
Basic structures
Principal ideal domains (PIDs)
Graded rings
Miscellaneous results
Chinese Remainder Theorem
Least common multiples