1 Algebraic Structures Binary operations defined on a set 2 Semigroups, Monoids, and Groups Definitions of semigroups, monoids, and groups 3 Grothendieck Group The Grothendieck group and the definition of integers 4 Group Homomorphisms Definitions and properties of group homomorphisms, their kernels and images 5 Quotient Groups Normal subgroups and quotient groups 6 Group Isomorphisms Isomorphism theorems 7 Direct Product of Groups Direct product of groups 8 Restricted Sum Restricted sums of groups 9 Free Product Free product and universal property 10 Abelian Groups Free abelian groups and tensor products 11 Group Action Group action 12 Definition of a Ring Definition and basic properties of rings 13 Quotient Rings and Ring Isomorphisms Quotient rings and ring isomorphism theorems 14 Products, Coproducts, and Tensor Products of Rings Definitions of products, coproducts, and tensor products of rings 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 a ring homomorphism 20 Graded Modules Definition of graded modules over a graded ring 21 Algebras Definition and types of algebras over commutative rings: associative, unital, and commutative 22 Direct Product, Direct Sum, and Tensor Product of Algebras Product, direct sum, and tensor product structures of algebras