Algebraic Structures
Binary operations defined on sets
Algebraic Structures
Semigroups, Monoids, Groups
Definitions of semigroups, monoids, and groups
Grothendieck Groups
The Grothendieck group and the definition of integers
Group Homomorphisms
Definition and properties of group homomorphisms, kernels and images
Quotient Groups
Normal subgroups and quotient groups
Isomorphism Theorems
The isomorphism theorems
Direct Products of Groups
Direct product of groups
Restricted Sums
Restricted sum of groups
Free Products
Free product and universal property
Abelian Groups
Free abelian group, tensor product
Group Actions
Group action
Definition of Rings
Definition and basic properties of rings
Quotient Rings and Ring Homomorphisms
Quotient ring and ring isomorphism theorems
Field of Fractions
Localization, ring of fraction, prime ideal
Modules
Definition of module