Algebraic Structures
Binary operations defined on sets
Algebraic Structures
Semigroups, Monoids, and 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
Isomorphism theorems
Direct Product of Groups
Direct product of groups
Restricted Sums
Restricted sums of groups
Free Products
Free product and universal property
Abelian Groups
Free abelian group, tensor product
Group Actions
Group action
Definition of a Ring
The definition of a ring and its basic properties
Quotient Rings and Ring Homomorphisms
Quotient ring and ring isomorphism theorems
Products, Coproducts, and Tensor Products of Rings
Definition of the product, coproduct, and tensor product of rings
Field of Fractions
Localization, ring of fractions, prime ideal
Graded Rings
Definition and basic properties of a graded ring indexed by a monoid
Modules
Definition of modules
Direct Products, Direct Sums, and Tensor Products of Modules
Product, coproduct, and tensor product in the module category
Change of Base Ring
Restriction and extension of scalars via a ring homomorphism
Graded Modules
The definition of graded modules over a graded ring