Ring Ontology

FLOW ALGEBRA
TITLE: Ring Ontology
DOMAIN: Discrete Mathematics / Algebraic Structures / Ring Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/algebraic-structures/ring-ontology-2.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_STATUS: Formal mathematical synthesis; live page not independently verified.

1. PURPOSE
Represent rings as typed algebraic structures with explicitly defined operations, axioms, substructures, morphisms, inference rules and machine-checkable constraints. Distinguish properties that depend on commutativity or the existence of a multiplicative identity.

2. TYPE SYSTEM
TYPE Set
TYPE Element
TYPE BinaryOperation: Set × Set -> Set
TYPE UnaryOperation: Set -> Set
TYPE Ring: (carrier:Set, add:BinaryOperation, multiply:BinaryOperation, zero:Element, negate:UnaryOperation, has_one:Bool)
TYPE Subring
TYPE Ideal
TYPE RingHomomorphism
TYPE QuotientRing
TYPE PolynomialRing
TYPE Boolean
TYPE ProofCertificate

3. FOUNDATIONAL DEFINITION
Ring(R) iff:
  (R,+) is an abelian group;
  multiplication is associative;
  multiplication distributes over addition on both sides.
FOR ALL a,b,c IN R:
  (a+b)+c = a+(b+c)
  a+b = b+a
  a+0 = a = 0+a
  a+(-a) = 0
  (ab)c = a(bc)
  a(b+c) = ab+ac
  (a+b)c = ac+bc
OPTIONAL UNIT AXIOM: EXISTS 1 IN R such that 1a=a=a1 for all a.
CONVENTION: The base definition permits rings without 1. Unital rings are marked explicitly.

4. DERIVED OPERATIONS AND LAWS
SUBTRACTION(a,b) := a+(-b)
ZERO_MULTIPLICATION: 0a = a0 = 0
NEGATION: (-a)b = -(ab) = a(-b)
DOUBLE_NEGATION: -(-a)=a
NEGATIVE_PRODUCT: (-a)(-b)=ab
POWER(a,0)=1 only for a unital ring; POWER(a,n+1)=POWER(a,n)a.

5. RING CLASSIFICATION
COMMUTATIVE_RING(R) iff Ring(R) and ab=ba for all a,b.
UNITAL_RING(R) iff Ring(R) and R has a multiplicative identity.
ZERO_RING(R) iff R={0}; its identity is 0 when a unit is specified.
INTEGRAL_DOMAIN(R) iff R is a nonzero commutative unital ring and ab=0 implies a=0 or b=0.
DIVISION_RING(R) iff R is nonzero, unital and each nonzero element is multiplicatively invertible.
FIELD(R) iff R is a commutative division ring.
ZERO_DIVISOR(a) iff a!=0 and EXISTS b!=0 with ab=0 or ba=0 (side-specific distinctions may be retained).
UNIT(a) iff EXISTS b with ab=ba=1 in a unital ring.
IDEMPOTENT(a) iff a*a=a.
NILPOTENT(a) iff EXISTS positive n with a^n=0.

6. STRUCTURAL RELATIONSHIPS
Ring IS_A AlgebraicStructure
Ring HAS_OPERATION Addition
Ring HAS_OPERATION Multiplication
Ring HAS_ADDITIVE_IDENTITY Zero
Ring HAS_ADDITIVE_INVERSE Negation
CommutativeRing SUBTYPE_OF Ring
UnitalRing SUBTYPE_OF Ring
IntegralDomain SUBTYPE_OF CommutativeUnitalRing
Field SUBTYPE_OF IntegralDomain
Ideal SUBSTRUCTURE_OF Ring
QuotientRing CONSTRUCTED_FROM (Ring, Ideal)
RingHomomorphism MAPS Ring TO Ring

7. HOMOMORPHISMS
RING_HOM(f:R->S) iff:
  f(a+b)=f(a)+f(b)
  f(ab)=f(a)f(b)
  f(0_R)=0_S
For unital homomorphisms, additionally require f(1_R)=1_S.
KERNEL(f)={a IN R | f(a)=0_S}; kernel is a two-sided ideal.
IMAGE(f)={f(a) | a IN R}; image is a subring.
FIRST_ISOMORPHISM_THEOREM: R/ker(f) is isomorphic to im(f), with compatible homomorphism conventions.

8. IDEALS AND QUOTIENTS
TWO_SIDED_IDEAL(I,R) iff I is an additive subgroup and
  FOR ALL r IN R, x IN I: rx IN I and xr IN I.
QUOTIENT(R,I) defined for two-sided ideals I.
COSSET_ADD: (a+I)+(b+I)=(a+b)+I
COSSET_MULTIPLY: (a+I)(b+I)=ab+I
VALIDATE quotient operations are independent of representative choice.

9. EXEMPLARS
INTEGER_RING: (Z,+,×,0,1) is a commutative unital ring and integral domain.
MODULAR_RING: Z/nZ for n>=2 is a commutative unital ring; a field iff n is prime.
MATRIX_RING: M_n(F), n>=2 over a field F, is unital and generally noncommutative.
POLYNOMIAL_RING: R[x] is a ring; commutative when R is commutative.
EVEN_INTEGERS: (2Z,+,×) is a commutative ring without multiplicative identity.

10. INFERENCE RULES
RULE R1: Ring(R) -> AbelianGroup(R,+).
RULE R2: Ring(R) -> Associative(R,×).
RULE R3: Ring(R) -> LeftDistributive(R,×,+) AND RightDistributive(R,×,+).
RULE R4: Ring(R) AND CommutativeMultiplication(R) -> CommutativeRing(R).
RULE R5: IntegralDomain(R) AND Finite(R) -> Field(R).
RULE R6: Field(R) -> IntegralDomain(R).
RULE R7: RingHomomorphism(f) -> KernelIsIdeal(f).
RULE R8: TwoSidedIdeal(I,R) -> WellDefinedQuotientRing(R,I).
RULE R9: Ring(R) AND a,b IN R -> a*0=0 AND 0*b=0.
Do not infer commutative multiplication, a multiplicative unit, or cancellation from Ring(R) alone.

11. VALIDATION MODEL
CHECK carrier nonempty and all operations closed.
CHECK additive associativity, commutativity, identity and inverses.
CHECK multiplicative associativity and both distributive laws.
IF has_one: CHECK two-sided multiplicative identity.
IF is_commutative: CHECK ab=ba.
IF integral_domain: CHECK nonzero, commutative, unital and no nonzero zero divisors.
IF quotient: CHECK ideal is two-sided and induced operations well-defined.
FINITE_RING: exhaustive validation possible by enumerating carrier tuples.
INFINITE_RING: require symbolic proofs or trusted theorem references; finite sampling is not proof.

12. SEMANTIC_EQUIVALENCE
'Ring' == 'Associative ring' under the explicitly declared convention of associative multiplication.
'Commutative ring' == 'Ring with commutative multiplication'.
'Ring with unity' == 'Unital ring'.
'Ring homomorphism' may or may not require unit preservation: specify convention.
NOT_EQUIVALENT: Ring != Field; Ring != IntegralDomain; Ideal != Subring.

13. CONFIDENCE
DEFINITIONAL_AXIOMS: 1.00 (mathematical certainty within declared definitions)
PROVEN_THEOREMS: 1.00 conditional on valid hypotheses and proof
INSTANCE_CLASSIFICATION: requires proof certificate or verified computation
UNCERTAIN_SOURCE_ALIGNMENT: live page not checked; do not assert verbatim correspondence

14. FLOW_HASH
CANONICALISE: Unicode NFC; normalise whitespace; preserve typed symbols and operator order.
HASH_ALGORITHM: SHA-256
HASH_INPUT: canonicalised flow payload excluding any embedded FLOW_HASH digest.
FLOW_HASH_VALUE: COMPUTE_AT_INGESTION (not fabricated).

15. UPDATE_MODEL
ON_CHANGE: increment version; record timestamp, author, changed axioms, dependencies and proof obligations.
REVALIDATE: all derived laws, ring classifications, quotient constructions and morphism constraints.
DEPENDENCIES: Group Ontology; Abelian Group Ontology; Semigroup Ontology; Field Ontology; Ideal Ontology.

16. AI_CONSUMPTION
ENTITY_EXTRACTION: Ring, Element, Operation, Ideal, Homomorphism, QuotientRing.
RELATION_EXTRACTION: HAS_OPERATION, SUBTYPE_OF, IS_A, MAPS_TO, CONSTRUCTED_FROM.
REASONING_MODE: typed, axiom-constrained symbolic inference.
SAFETY: reject invalid algebraic generalisations and missing preconditions.
OUTPUT: definitions, structured relations, proof obligations, validation outcomes and provenance.

END FLOW ALGEBRA