This chapter defines algebraic structures, starting with internal composition laws on a set G, illustrated by addition and multiplication on number sets and composition of maps.  Associative, commutative, neutral, and inverse elements are defined, leading to the definition of a group (G,⋆) requiring associativity, a neutral element, and inverses for all elements.  Commutative groups are also defined.  Key theorems establish the uniqueness of the neutral and inverse elements, and a formula for the inverse of a product.  Subgroups H of G are defined, characterized by non-emptiness, closure under the operation and inverses.  Equivalent conditions for subgroups are provided, with specific examples for additive and multiplicative groups.  The chapter then introduces rings (A,+,.), defined as a commutative group under (+) and an associative operation (∙) distributive over (+). Commutative rings and rings with identity are defined. Subrings are defined and characterized. Finally, fields (�,+,.) are defined as rings where all non-zero elements are invertible; commutative fields are also defined. Subfields are characterized as subrings where inverses of non-zero elements exist, with examples showing (Q,+,.) and (R,+,.) as subfields of (C,+,.).