ring

Type Members

1. trait AdditiveCommutativeGroup[A] extends AdditiveGroup[A] with AdditiveCommutativeMonoid[A]

   

2. trait AdditiveCommutativeMonoid[A] extends AdditiveMonoid[A] with AdditiveCommutativeSemigroup[A]

   

3. trait AdditiveCommutativeSemigroup[A] extends AdditiveSemigroup[A]

   

4. trait AdditiveGroup[A] extends AdditiveMonoid[A]

   

5. trait AdditiveGroupFunctions[G[T] <: AdditiveGroup[T]] extends AdditiveMonoidFunctions[G]

   

6. trait AdditiveMonoid[A] extends AdditiveSemigroup[A]

   

7. trait AdditiveMonoidFunctions[M[T] <: AdditiveMonoid[T]] extends AdditiveSemigroupFunctions[M]

   

8. trait AdditiveSemigroup[A] extends Serializable

   

9. trait AdditiveSemigroupFunctions[S[T] <: AdditiveSemigroup[T]] extends AnyRef

   

10. trait BoolRing[A] extends BoolRng[A] with CommutativeRing[A]

    

    A Boolean ring is a ring whose multiplication is idempotent, that is a⋅a = a for all elements a.

    A Boolean ring is a ring whose multiplication is idempotent, that is a⋅a = a for all elements a. This property also implies a+a = 0 for all a, and a⋅b = b⋅a (commutativity of multiplication).

    Every Boolean ring is equivalent to a Boolean algebra. See algebra.lattice.BoolFromBoolRing for details.

11. trait BoolRng[A] extends Rng[A]

    

    A Boolean rng is a rng whose multiplication is idempotent, that is a⋅a = a for all elements a.

    A Boolean rng is a rng whose multiplication is idempotent, that is a⋅a = a for all elements a. This property also implies a+a = 0 for all a, and a⋅b = b⋅a (commutativity of multiplication).

    Every BoolRng is equivalent to algebra.lattice.GenBool. See algebra.lattice.GenBoolFromBoolRng for details.

12. trait CommutativeRig[A] extends Rig[A] with MultiplicativeCommutativeMonoid[A]

    

    CommutativeRig is a Rig that is commutative under multiplication.

13. trait CommutativeRing[A] extends Ring[A] with CommutativeRig[A]

    

    CommutativeRing is a Ring that is commutative under multiplication.

14. trait EuclideanRing[A] extends CommutativeRing[A]

    

    EuclideanRing implements a Euclidean domain.

    EuclideanRing implements a Euclidean domain.

    The formal definition says that every euclidean domain A has (at least one) euclidean function f: A -> N (the natural numbers) where:

    (for every x and non-zero y) x = yq + r, and r = 0 or f(r) < f(y).

    The idea is that f represents a measure of length (or absolute value), and the previous equation represents finding the quotient and remainder of x and y. So:

    quot(x, y) = q mod(x, y) = r

    This type does not provide access to the Euclidean function, but only provides the quot, mod, and quotmod operators.

15. trait EuclideanRingFunctions[R[T] <: EuclideanRing[T]] extends RingFunctions[R]

    

16. trait Field[A] extends EuclideanRing[A] with MultiplicativeCommutativeGroup[A]

    

17. trait FieldFunctions[F[T] <: Field[T]] extends EuclideanRingFunctions[F] with MultiplicativeGroupFunctions[F]

    

18. trait MultiplicativeCommutativeGroup[A] extends MultiplicativeGroup[A] with MultiplicativeCommutativeMonoid[A]

    

19. trait MultiplicativeCommutativeMonoid[A] extends MultiplicativeMonoid[A] with MultiplicativeCommutativeSemigroup[A]

    

20. trait MultiplicativeCommutativeSemigroup[A] extends MultiplicativeSemigroup[A]

    

21. trait MultiplicativeGroup[A] extends MultiplicativeMonoid[A]

    

22. trait MultiplicativeGroupFunctions[G[T] <: MultiplicativeGroup[T]] extends MultiplicativeMonoidFunctions[G]

    

23. trait MultiplicativeMonoid[A] extends MultiplicativeSemigroup[A]

    

24. trait MultiplicativeMonoidFunctions[M[T] <: MultiplicativeMonoid[T]] extends MultiplicativeSemigroupFunctions[M]

    

25. trait MultiplicativeSemigroup[A] extends Serializable

    

26. trait MultiplicativeSemigroupFunctions[S[T] <: MultiplicativeSemigroup[T]] extends AnyRef

    

27. trait Rig[A] extends Semiring[A] with MultiplicativeMonoid[A]

    

    Rig consists of:

    Rig consists of:

    • a commutative monoid for addition (+)
    • a monoid for multiplication (*)

    Alternately, a Rig can be thought of as a ring without multiplicative or additive inverses (or as a semiring with a multiplicative identity).

    Mnemonic: "Rig is a Ring without 'N'egation."

28. trait Ring[A] extends Rig[A] with Rng[A]

    

    Ring consists of:

    Ring consists of:

    • a commutative group for addition (+)
    • a monoid for multiplication (*)

    Additionally, multiplication must distribute over addition.

    Ring implements some methods (for example fromInt) in terms of other more fundamental methods (zero, one and plus). Where possible, these methods should be overridden by more efficient implementations.

29. trait RingFunctions[R[T] <: Ring[T]] extends AdditiveGroupFunctions[R] with MultiplicativeMonoidFunctions[R]

    

30. trait Rng[A] extends Semiring[A] with AdditiveCommutativeGroup[A]

    

    Rng (pronounced "Rung") consists of:

    Rng (pronounced "Rung") consists of:

    • a commutative group for addition (+)
    • a semigroup for multiplication (*)

    Alternately, a Rng can be thought of as a ring without a multiplicative identity (or as a semiring with an additive inverse).

    Mnemonic: "Rng is a Ring without multiplicative 'I'dentity."

31. trait Semiring[A] extends AdditiveCommutativeMonoid[A] with MultiplicativeSemigroup[A]

    

    Semiring consists of:

    Semiring consists of:

    • a commutative monoid for addition (+)
    • a semigroup for multiplication (*)

    Alternately, a Semiring can be thought of as a ring without a multiplicative identity or an additive inverse.

    A Semiring with an additive inverse (-) is a Rng. A Semiring with a multiplicative identity (1) is a Rig. A Semiring with both of those is a Ring.