1 Algebraic Structures Binary operations defined on a set 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 Definitions and properties of group homomorphisms, kernels and images 5 Quotient Groups Normal subgroups and quotient groups 6 Group Homomorphisms Homomorphism theorems 7 Direct Product of Groups Categorical definition and universal property of group products 8 Restricted Direct Sum Restricted sums of groups 9 Free Products Free products and the universal property 10 Abelian Groups Free abelian groups and tensor products 11 Group Actions Group actions on sets 12 Definition of a Ring Definition and basic properties of rings 13 Quotient Rings and Ring Homomorphisms Quotient rings and ring isomorphism theorems 14 Products, Coproducts, and Tensor Products of Rings Categorical definitions of ring products, coproducts, and tensor products 15 Field of Fractions Localization, ring of fractions, prime ideal 16 Graded Rings Definition and basic properties of monoid-indexed graded rings 17 Modules Definition of a module 18 Direct Products, Direct Sums, and Tensor Products of Modules Products, coproducts, and tensor products in the module category 19 Change of Scalars Restriction and extension of scalars via ring homomorphism 20 Graded Modules Definition of graded modules over a graded ring 21 Algebras Definitions and types of algebras over commutative rings 22 Direct Product, Direct Sum, and Tensor Product of Algebras Product, direct sum, and tensor product structures of algebras