Statements (89)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:algebra
|
| gptkbp:additionHasIdentity |
yes
|
| gptkbp:additionHasInverses |
yes
|
| gptkbp:additionIs |
associative
commutative |
| gptkbp:defines |
set equipped with two binary operations satisfying certain axioms
|
| gptkbp:distributiveLaw |
multiplication distributes over addition
|
| gptkbp:example |
polynomial ring
integers matrix ring function ring modular arithmetic ring |
| gptkbp:generalizes |
gptkb:commutative_ring
gptkb:Field gptkb:Boolean_ring gptkb:integral_domain division ring matrix ring |
| gptkbp:hasApplication |
gptkb:combinatorics
gptkb:algebraic_geometry coding theory computer science cryptography number theory physics |
| gptkbp:hasProperty |
associativity
distributivity commutativity of addition zero element additive inverse may be commutative may be noncommutative may be without zero divisors may have unity may have zero divisors may lack unity multiplicative identity (optional) multiplicative inverse (not required) |
| gptkbp:introduced |
gptkb:Richard_Dedekind
|
| gptkbp:introducedIn |
gptkb:19th_century
|
| gptkbp:multiplicationHasIdentity |
not required
|
| gptkbp:multiplicationIs |
associative
not necessarily commutative |
| gptkbp:operator |
addition
multiplication |
| gptkbp:relatedConcept |
gptkb:Euclidean_domain
gptkb:group_of_people gptkb:model gptkb:Noetherian_ring gptkb:commutative_ring gptkb:local_ring gptkb:Field gptkb:Artinian_ring gptkb:C*-algebra gptkb:Jacobson_radical gptkb:Boolean_ring gptkb:von_Neumann_regular_ring gptkb:semiring gptkb:topological_ring gptkb:integral_domain gptkb:principal_ideal_domain gptkb:unique_factorization_domain polynomial ring power series ring algebra over a field commutator endomorphism ring ring homomorphism automorphism group center of a ring division ring semisimple ring simple ring group ring direct product of rings matrix ring nilradical Banach ring direct sum of rings tensor product of rings |
| gptkbp:studiedIn |
gptkb:algebra
abstract algebra |
| gptkbp:subunit |
gptkb:ideal
ring homomorphism quotient ring subring |
| gptkbp:bfsParent |
gptkb:Left_ideal
|
| gptkbp:bfsLayer |
7
|
| https://www.w3.org/2000/01/rdf-schema#label |
Ring (mathematics)
|