1 Algebraic Structures Binary operations defined on sets 2 Semigroups, Monoids, and Groups Definitions of semigroups, monoids, and groups 3 Grothendieck Groups The Grothendieck group and the definition of integers 4 Group Homomorphisms Definition and properties of group homomorphisms, kernels and images 5 Quotient Groups Normal subgroups and quotient groups 6 Isomorphism Theorems Isomorphism theorems 7 Direct Product of Groups Direct product of groups 8 Restricted Sums Restricted sums of groups 9 Free Products Free product and universal property 10 Abelian Groups Free abelian group, tensor product 11 Group Actions Group action 12 Definition of a Ring The definition of a ring and its basic properties 13 Quotient Rings and Ring Homomorphisms Quotient ring and ring isomorphism theorems 14 Products, Coproducts, and Tensor Products of Rings Definition of the product, coproduct, and tensor product of rings 15 Field of Fractions Localization, ring of fractions, prime ideal 16 Graded Rings Definition and basic properties of a graded ring indexed by a monoid 17 Modules Definition of modules 18 Direct Products, Direct Sums, and Tensor Products of Modules Products, coproducts, and tensor products in the module category 19 Change of Base Ring Restriction and extension of scalars via a ring homomorphism 20 Graded Modules The definition of graded modules over a graded ring 21 Algebras The definition of an algebra over a commutative ring, and various kinds such as associative, unital, and co...