graph TD

            %% CANCELLATION LAW (Applies to multiple areas)
            CL["Cancellation Law
(Property allowing elements to be canceled in equations)"]

            %% GROUP THEORY
            subgraph GroupTheory [Group Theory]
                G["Groups
(Set with associative binary operation, identity, & inverses)"]
                AG["Abelian Groups
(Groups where the operation is commutative)"]
                CG["Cyclic Groups
(Groups generated by a single element)"]
                SubG["Subgroups
(Subset that forms a group under the same operation)"]
                NG["Normal Subgroups
(Subgroup invariant under conjugation)"]
                QG["Quotient Groups
(Group formed by cosets of a normal subgroup)"]
                ComS["Commutator Subgroup
(Generated by all commutators; measures non-abelianness)"]
                GP["Group Products
(Constructing a new group from the Cartesian product)"]
                Ord["Orders
(Number of elements in a group, or generated by an element)"]
                CP["Cyclic Permutations
(Permutations that shift elements in a single cycle)"]

                G -->|has special type| AG
                G -->|has special type| CG
                G -->|contains| SubG
                G -->|combined via| GP
                G -->|size measured by| Ord
                SubG -->|special type| NG
                NG -->|used to construct| QG
                G -->|contains| ComS
                CG -->|example elements| CP
                G -.->|holds by definition| CL
            end

            %% RING THEORY
            subgraph RingTheory [Ring Theory]
                Dist["Distributivity
(Axiom connecting addition and multiplication)"]
                R["Rings
(Set acting as an Abelian group for addition & associative for multiplication)"]
                CR["Commutative Rings
(Rings where multiplication is commutative)"]
                SubR["Subrings
(Subset forming a ring)"]
                Uni["Unity
(Multiplicative identity element)"]
                ID["Integral Domains
(Commutative ring with unity and no zero divisors)"]
                Char["Characteristic
(Smallest integer n such that n*1=0)"]
                PI["Principle Ideals
(Ideal generated by a single element)"]
                MI["Maximal Ideals
(Proper ideal not contained in any other proper ideal)"]

                AG -->|adds multiplication and| Dist
                Dist -->|defines| R
                R -->|if commutative| CR
                R -->|contains| SubR
                R -->|may possess| Uni
                R -->|property| Char
                CR -->|if unity & no zero divisors| ID
                SubR -->|absorbing subsets act as| PI
                PI -->|can be strictly| MI
                ID -.->|holds for non-zero elements| CL
            end

            %% FIELD THEORY
            subgraph FieldTheory [Field Theory]
                F["Fields
(Commutative ring with unity where every non-zero element has an inverse)"]
                EF["Extension Fields
(A larger field containing a base field)"]
                VS["Vector Spaces
(Structure with vector addition & scalar multiplication over a field)"]
                PR["Polynomial Rings
(Ring of polynomials formed over a base field or ring)"]
                Red["Reducibility
(Whether a polynomial can be factored)"]
                EC["Eisenstein Criterion
(Test for polynomial irreducibility over a field)"]
                Trans["Transcendentals
(Elements in an extension not the root of any base polynomial)"]
                FLT["Fermat's Little Theorem
(a^p = a mod p; applies to finite fields)"]

                ID -->|if every non-zero element is invertible| F
                MI -->|quotient ring by maximal ideal creates| F
                F -->|can be extended to| EF
                F -->|provides scalars for| VS
                F -->|provides coefficients for| PR
                PR -->|property of polynomials| Red
                Red -->|tested by| EC
                EF -->|contains algebraic roots or| Trans
                F -.->|theorem related to prime fields| FLT
            end

            %% MORPHISMS & MAPPINGS
            subgraph Morphisms [Morphisms & Mappings]
                Hom["Homomorphism
(Structure-preserving map between algebraic structures)"]
                Iso["Isomorphisms
(Bijective homomorphism; structures are essentially identical)"]
                Auto["Automorphisms
(Isomorphism from a structure to itself)"]
                RH["Ring Homomorphism
(Preserves both addition and multiplication)"]
                FH["Field Homomorphisms
(Preserves field structure; always injective)"]

                Hom -->|if bijective| Iso
                Iso -->|if domain = codomain| Auto
                Hom -->|applied to rings| RH
                Hom -->|applied to fields| FH
                
                %% Connect to main areas
                G -.->|studied via| Hom
                R -.->|studied via| RH
                F -.->|studied via| FH
            end