Ring Ontology

FLOW Ring_Ontology_v1.0

PAGE:
  title: "Ring Ontology"
  canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/ring-ontology.html"
  language: "en-GB"
  domain: "Mathematics > Algebra > Abstract Algebra > Rings"

DEFINE RingOntology := {

  CORE_IDENTITY: {
    entity_type: "Mathematical Ontology",
    primary_entity: Ring,
    parent_entities: [AlgebraicStructure, SetWithOperations],
    related_entities: [
      Set,
      Addition,
      Multiplication,
      AdditiveIdentity,
      AdditiveInverse,
      MultiplicativeIdentity,
      Commutativity,
      Associativity,
      Distributivity,
      IntegralDomain,
      Field,
      Ideal,
      QuotientRing,
      Subring,
      RingHomomorphism
    ]
  },

  DEFINITION: {
    meaning:
      "A ring is a set equipped with addition and multiplication such that addition forms an abelian group, multiplication is associative, and multiplication distributes over addition.",
    structure:
      (R,+,*),
    common_unital_convention:
      may_require_multiplicative_identity_1,
    ontology_policy:
      explicitly_record_whether_identity_is_required_in_context
  },

  SEMANTIC_PRIMITIVES: {
    CarrierSet:
      R,
    Addition:
      binary_operation_on_R,
    Multiplication:
      binary_operation_on_R,
    Zero:
      additive_identity,
    AdditiveInverse:
      -a,
    One:
      multiplicative_identity_when_present
  },

  ADDITIVE_STRUCTURE: {
    closure:
      a+b IN R,

    associativity:
      (a+b)+c=a+(b+c),

    identity:
      a+0=a,

    inverse:
      a+(-a)=0,

    commutativity:
      a+b=b+a
  },

  MULTIPLICATIVE_STRUCTURE: {
    closure:
      a*b IN R,

    associativity:
      (a*b)*c=a*(b*c),

    identity_optional_by_definition:
      1*a=a*1=a
      WHEN ring_is_unital,

    commutativity_optional:
      a*b=b*a
      only_in_commutative_ring
  },

  DISTRIBUTIVE_LAWS: {
    left:
      a*(b+c)=a*b+a*c,

    right:
      (a+b)*c=a*c+b*c
  },

  CLASSIFICATION: {

    UnitalRing: {
      has_multiplicative_identity:
        TRUE
    },

    NonUnitalRing: {
      has_multiplicative_identity:
        FALSE
    },

    CommutativeRing: {
      multiplication_commutative:
        TRUE
    },

    NoncommutativeRing: {
      multiplication_commutative:
        FALSE
    },

    IntegralDomain: {
      commutative:
        TRUE,
      unital:
        TRUE,
      zero_divisors:
        NONE
    },

    DivisionRing: {
      every_nonzero_element_has_multiplicative_inverse:
        TRUE,
      multiplication_commutative:
        not_required
    },

    Field: {
      DivisionRing:
        TRUE,
      multiplication_commutative:
        TRUE
    }
  },

  ZERO_DIVISORS: {
    definition:
      nonzero_a_and_b_with_a*b=0,

    ring_with_zero_divisors:
      may_fail_cancellation,

    integral_domain:
      has_no_zero_divisors
  },

  NILPOTENT_ELEMENT: {
    definition:
      a^n=0
      for_some_positive_integer_n,

    semantic_note:
      nonzero_nilpotent_elements_are_zero_divisors_in_commutative_rings
  },

  IDEMPOTENT_ELEMENT: {
    definition:
      e^2=e
  },

  UNIT_ELEMENT: {
    definition:
      u_has_multiplicative_inverse_v
      SUCH_THAT u*v=v*u=1,

    set_of_units:
      R^*
  },

  SUBRING: {
    definition:
      S SUBSET_OR_EQUAL R
      AND S_is_ring_under_inherited_operations,

    common_test:
      nonempty
      AND closed_under_subtraction
      AND closed_under_multiplication
  },

  IDEAL: {
    additive_subgroup:
      TRUE,

    absorption:
      FOR_ALL r IN R, a IN I:
        r*a IN I
        AND
        a*r IN I
      for_two_sided_ideal,

    variants: [
      left_ideal,
      right_ideal,
      two_sided_ideal
    ]
  },

  PRINCIPAL_IDEAL: {
    generated_by_a:
      (a),

    commutative_ring:
      (a)={r*a | r IN R}
  },

  QUOTIENT_RING: {
    construction:
      R/I,

    condition:
      I_is_two_sided_ideal,

    elements:
      cosets_r+I,

    operations:
      induced_addition_and_multiplication
  },

  RING_HOMOMORPHISM: {
    phi:R->S,

    preserves_addition:
      phi(a+b)=phi(a)+phi(b),

    preserves_multiplication:
      phi(a*b)=phi(a)*phi(b),

    unital_context:
      often_require_phi(1_R)=1_S
  },

  KERNEL: {
    ker(phi):
      {r IN R | phi(r)=0},

    property:
      ideal_of_R
  },

  IMAGE: {
    im(phi):
      {phi(r) | r IN R},

    property:
      subring_of_S
  },

  FIRST_ISOMORPHISM_THEOREM: {
    relation:
      R/ker(phi) ≅ im(phi)
      under_standard_hypotheses
  },

  RING_ISOMORPHISM: {
    definition:
      bijective_ring_homomorphism,

    consequence:
      preserves_ring_structure
  },

  POLYNOMIAL_RING: {
    notation:
      R[x],

    coefficients:
      in_R,

    structure:
      ring_under_polynomial_addition_and_multiplication
  },

  MATRIX_RING: {
    notation:
      M_n(R),

    elements:
      n_by_n_matrices_over_R,

    multiplication:
      matrix_multiplication,

    commutativity:
      generally_FALSE_for_n>1
  },

  INTEGER_RING: {
    Integers:
      commutative_unital_ring,

    also:
      integral_domain
  },

  MODULAR_RING: {
    Z/nZ:
      finite_commutative_unital_ring,

    field_condition:
      n_is_prime
  },

  PRODUCT_RING: {
    R×S:
      componentwise_addition_and_multiplication
  },

  CHARACTERISTIC: {
    definition:
      least_positive_n_with_n*1=0
      for_unital_ring,

    IF none:
      characteristic=0
  },

  CANCELLATION: {
    additive:
      always_valid,

    multiplicative:
      not_valid_in_general,

    valid_if:
      cancelling_element_not_zero_divisor
  },

  DISTRIBUTIVE_DERIVATIONS: {
    zero_multiplication:
      a*0=0=0*a,

    negative_multiplication:
      (-a)*b=-(a*b),

    double_negative:
      (-a)*(-b)=a*b
  },

  CENTER_OF_RING: {
    Z(R):
      {z IN R | z*r=r*z FOR_ALL r IN R},

    property:
      commutative_subring
  },

  COMMUTATOR: {
    [a,b]:
      a*b-b*a,

    commutative_ring:
      commutator_always_zero
  },

  OPPOSITE_RING: {
    R^op:
      same_additive_group,
      reversed_multiplication_order
  },

  MODULE_RELATION: {
    ring:
      provides_scalar_structure_for_modules,

    analogy:
      modules_generalise_vector_spaces_from_fields_to_rings
  },

  ALGEBRA_RELATION: {
    algebra_over_field_or_commutative_ring:
      ring_with_compatible_scalar_multiplication
  },

  FACTORISATION_RELATION: {
    integral_domains:
      support_divisibility_theory,

    stronger_classes: [
      UFD,
      PID,
      EuclideanDomain
    ]
  },

  EUCLIDEAN_DOMAIN_RELATION: {
    EuclideanDomain:
      integral_domain_with_Euclidean_function,

    consequence:
      PID
  },

  PID_RELATION: {
    PrincipalIdealDomain:
      integral_domain_where_every_ideal_is_principal,

    consequence:
      UFD
  },

  UFD_RELATION: {
    UniqueFactorisationDomain:
      integral_domain_with_unique_factorisation_into_irreducibles_up_to_units_and_order
  },

  PRIME_ELEMENT: {
    in_commutative_integral_domain:
      p_divides_a*b
      -> p_divides_a OR p_divides_b
  },

  IRREDUCIBLE_ELEMENT: {
    nonzero_nonunit:
      cannot_factor_into_two_nonunits
  },

  PRIME_IDEAL: {
    commutative_ring_R:

    proper_ideal_P:
      IF a*b IN P
      THEN a IN P OR b IN P
  },

  MAXIMAL_IDEAL: {
    proper_ideal_M:
      no_proper_ideal_strictly_between_M_and_R
  },

  QUOTIENT_CHARACTERISATIONS: {
    R/P_integral_domain:
      IFF P_prime
      for_commutative_ring_with_identity,

    R/M_field:
      IFF M_maximal
  },

  BOOLEAN_RING: {
    property:
      x^2=x FOR_ALL x,

    consequence:
      commutative
      AND characteristic_2
  },

  ZERO_RING: {
    condition:
      0=1,

    convention_note:
      some_definitions_allow_zero_ring,
      fields_exclude_zero_ring
  },

  SYMBOLIC_RECOGNITION: {
    ACCEPT: [
      "Z under + and ×",
      "Z/nZ",
      "M_2(R)",
      "R[x]",
      "Q[x]"
    ],

    CLASSIFY_MORE_SPECIFICALLY: [
      "Q -> field",
      "Z -> integral domain",
      "M_2(R) -> noncommutative ring"
    ]
  },

  NATURAL_LANGUAGE_MAPPING: {
    "ring":
      additive_abelian_group_plus_associative_distributive_multiplication,

    "commutative ring":
      ring_with_commutative_multiplication,

    "ideal":
      absorbent_additive_subgroup,

    "quotient ring":
      coset_ring_modulo_ideal
  },

  ERROR_DETECTION: {
    AdditiveGroupFailure:
      -> reject_ring,

    MultiplicationNonassociative:
      -> reject_ring_under_standard_definition,

    DistributivityFailure:
      -> reject_ring,

    IdentityConventionConfusion:
      ring_without_one_misclassified_under_unital_only_definition
      -> flag,

    FieldConfusion:
      ring_with_noninvertible_nonzero_elements_called_field
      -> flag,

    ZeroDivisorCancellationError:
      invalid_multiplicative_cancellation
      -> flag
  },

  ONTOLOGY_RELATIONS: {
    IS_A: [
      "CommutativeRing IS_A Ring",
      "IntegralDomain IS_A CommutativeRing",
      "DivisionRing IS_A Ring",
      "Field IS_A CommutativeRing"
    ],

    RELATED_TO: [
      "Ring RELATED_TO Ideal",
      "Ring RELATED_TO QuotientRing",
      "Ring RELATED_TO Module",
      "Ring RELATED_TO PolynomialRing",
      "Ring RELATED_TO MatrixRing",
      "Ring RELATED_TO Field"
    ],

    DEPENDS_ON: [
      Set,
      AbelianGroup,
      BinaryOperation,
      Associativity,
      Distributivity
    ]
  },

  SEMANTIC_EQUIVALENCE: {
    ring_isomorphism:
      structural_equivalence,

    quotient_equivalence:
      context_dependent_via_isomorphism_theorems
  },

  CONFIDENCE: {
    classification_confidence:
      IF additive_axioms
      AND multiplicative_associativity
      AND distributivity_verified
      THEN 1.0,

    commutativity_confidence:
      IF multiplication_commutativity_verified
      THEN 1.0,

    identity_confidence:
      IF multiplicative_identity_verified_or_explicitly_absent
      THEN 1.0
  },

  AI_CONSUMPTION: {
    tasks: [
      recognise_ring_candidate,
      verify_additive_abelian_group,
      verify_multiplication_closure,
      verify_multiplication_associativity,
      verify_distributivity,
      detect_identity,
      detect_commutativity,
      detect_zero_divisors,
      classify_integral_domain_division_ring_or_field,
      identify_subrings,
      identify_ideals,
      construct_quotient_rings,
      analyse_homomorphisms,
      link_factorisation_domains,
      detect_ring_axiom_failures
    ],

    preferred_representation:
      typed_algebraic_structure_with_operation_and_axiom_graph,

    reasoning_mode:
      recursive_axiom_and_substructure_analysis
  },

  RECURSIVE_REASONING: {

    FUNCTION AnalyseRingCandidate(S,Plus,Times,Context):

      carrier := ParseSet(S)
      add := ParseOperation(Plus)
      mult := ParseOperation(Times)

      IF not VerifyAbelianGroup(carrier,add):
        RETURN NotRing("additive structure invalid")

      IF not VerifyClosure(carrier,mult):
        RETURN NotRing("multiplication not closed")

      IF not VerifyAssociativity(carrier,mult):
        RETURN NotRing("multiplication not associative")

      IF not VerifyDistributivity(carrier,add,mult):
        RETURN NotRing("distributivity invalid")

      has_identity := DetectMultiplicativeIdentity(carrier,mult)
      commutative := VerifyCommutativity(carrier,mult)
      zero_divisors := DetectZeroDivisors(carrier,mult)

      classifications := [Ring]

      IF has_identity:
        classifications += UnitalRing

      IF commutative:
        classifications += CommutativeRing

      IF commutative AND has_identity AND zero_divisors=NONE:
        classifications += IntegralDomain

      IF has_identity
      AND EveryNonzeroElementInvertible(carrier,mult):
        classifications += DivisionRing

      IF DivisionRing IN classifications AND commutative:
        classifications += Field

      RETURN {
        entity:Ring,
        carrier,
        operations:[add,mult],
        classifications,
        has_identity,
        commutative,
        zero_divisors,
        verified:TRUE
      }
  },

  RECURSIVE_COHERENCE: {
    invariant_1:
      additive_structure_is_abelian_group=TRUE,

    invariant_2:
      multiplication_is_associative=TRUE,

    invariant_3:
      distributive_laws_hold=TRUE,

    invariant_4:
      field_classification_implies_ring_axioms=TRUE,

    invariant_5:
      quotient_ring_requires_ideal=TRUE
  },

  FLOW_HASH: {
    semantic_signature_inputs: [
      ontology_name,
      ring_definition,
      additive_axioms,
      multiplication_axioms,
      distributive_axioms,
      classification_model,
      ideal_and_quotient_relations
    ],

    purpose:
      "Detect semantic drift between revisions of the ontology."
  },

  UPDATE_MODEL: {
    version: "1.0",

    immutable_constraints: [
      additive_abelian_group,
      multiplication_associative,
      distributive_laws,
      zero_divisor_semantics,
      ideal_absorption_semantics
    ],

    revision_triggers: [
      semantic_conflict,
      GroupOntology_update,
      FieldOntology_update,
      IdealOntology_update,
      representation_standard_change
    ]
  },

  KNOWLEDGE_GRAPH: {
    nodes: [
      Ring,
      AlgebraicStructure,
      CarrierSet,
      Addition,
      Multiplication,
      Zero,
      One,
      ZeroDivisor,
      Unit,
      Subring,
      Ideal,
      QuotientRing,
      RingHomomorphism
    ],

    edges: [
      Ring -> IS_A -> AlgebraicStructure,
      Ring -> HAS_CARRIER -> CarrierSet,
      Ring -> HAS_OPERATION -> Addition,
      Ring -> HAS_OPERATION -> Multiplication,
      Addition -> HAS_IDENTITY -> Zero,
      UnitalRing -> HAS_IDENTITY -> One,
      Ring -> MAY_HAVE -> ZeroDivisor,
      Ring -> MAY_HAVE -> Unit,
      Subring -> SUBSET_OF -> Ring,
      Ring -> HAS_IDEAL -> Ideal,
      QuotientRing -> FORMED_FROM -> Ring,
      RingHomomorphism -> MAPS -> Ring
    ]
  },

  CROSS_ONTOLOGY_LINKS: {
    GroupOntology:
      additive_abelian_group_foundation,

    FieldOntology:
      stronger_commutative_division_structure,

    IntegralDomainOntology:
      no_zero_divisors_specialisation,

    ModuleOntology:
      scalar_ring_structure,

    PolynomialOntology:
      polynomial_rings,

    MatrixOntology:
      matrix_rings,

    IdealOntology:
      quotient_and_structure_theory,

    NumberTheoryOntology:
      modular_rings_and_factorisation
  },

  TERMINATION_CONDITION: {
    ontology_complete_when: [
      ring_defined,
      additive_structure_defined,
      multiplication_structure_defined,
      distributivity_defined,
      identity_convention_defined,
      ring_classes_defined,
      zero_divisors_and_units_defined,
      subrings_and_ideals_defined,
      quotient_rings_defined,
      homomorphisms_defined,
      error_detection_defined,
      cross_ontology_links_defined,
      recursive_coherence_verified
    ]
  }

}

EXECUTE Ring_Ontology_v1.0:

  INPUT:
    mathematical_entity_or_query

  PIPELINE:
    Parse
    -> IdentifyCarrierSetAndOperations
    -> VerifyAdditiveAbelianGroup
    -> VerifyMultiplicationClosure
    -> VerifyMultiplicationAssociativity
    -> VerifyDistributivity
    -> DetectMultiplicativeIdentity
    -> DetectCommutativity
    -> DetectZeroDivisorsAndUnits
    -> ClassifyRingType
    -> AnalyseSubringsIdealsAndQuotients
    -> AnalyseHomomorphismsWhenRequested
    -> VerifySemanticConsistency
    -> LinkRelatedOntologies
    -> AssignConfidence
    -> EmitSemanticResult

  OUTPUT: {
    entity,
    classification,
    carrier_set,
    addition_operation,
    multiplication_operation,
    additive_identity,
    multiplicative_identity,
    commutativity,
    zero_divisors,
    units,
    subrings,
    ideals,
    quotient_rings,
    homomorphisms,
    characteristic,
    verification,
    ontology_relations,
    confidence,
    semantic_equivalence,
    flow_hash
  }

END FLOW Ring_Ontology_v1.0