We are going to learn Abstract Algebra about this topic. Firs look, what are we are going to learn or our learning course content.

1.1 Defining Groups

1.2 Defining Rings

1.3 Examples

1.4 First Concepts

1.5 Ideals

1.6 Homomorphisms

1.7 Products

2 Commutative Rings

2.1 Maximal Ideals

2.2 Prime Ideals

2.3 Radical Ideals

2.4 Noetherian Rings

2.5 Unique Factorisation Domains

2.6 Principal Ideal Domains

2.7 Lasker-Noether Decomposition

2.8 Finite Rings

2.9 Localisation

2.10 Local Rings

2.11 Dedekind Domains

3 Modules

3.1 Defining Modules

3.2 First Concepts

3.3 Free Modules

3.4 Homomorphisms

3.5 Rank of Modules

3.6 Length of Modules

3.7 Localisation of Modules

4 Linear Algebra

4.1 Matices

4.2 Elementary Matrices

4.3 Linear Equations

4.4 Determinants

4.5 Rank of Matrices

4.6 Canonical Forms

5 Structure Theorems

5.1 Asociated Primes

5.2 Primary Decomposition

5.3 The Theorem of Prufer

6 Polynomial Rings

6.1 Monomial Orders

6.2 Graded Algebras

6.3 Defining Polynomials

6.4 The Standard Cases

6.5 Roots of Polynomials

6.6 Derivation of Polynomials

6.7 Algebraic Properties

6.8 Grobner Bases

6.9 Algorithms

7 Polynomials in One Variable 209

7.1 Interpolation

7.2 Irreducibility Tests

7.3 Symmetric Polynomials

7.4 The Resultant

7.5 The Discriminant

7.6 Polynomials of Low Degree

7.7 Polynomials of High Degree

7.8 Fundamental Theorem of Algebra

8 Group Theory

9 Multilinear Algebra

9.1 Multilinear Maps

9.2 Duality Theory

9.3 Tensor Product of Modules

9.4 Tensor Product of Algebras

9.5 Tensor Product of Maps

9.6 Di erentials

10 Categorial Algebra

10.1 Sets and Classes

10.2 Categories and Functors

10.3 Examples

10.4 Product

10.5 Abelian Categories

10.6 Homology

11 Ring Extensions

12 Galois Theory

13 Graded Rings

14 Valuations

15 Proofs - Fundamentals

16 Proofs - Rings and Modules 246

17 Proofs - Commutative Algebra 300

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