NumHask.Algebra.Ring

Ring classes. A distinguishment is made between Rings and Commutative Rings.

Documentation

class (MultiplicativeAssociative a, MultiplicativeUnital a, Distribution a) => Semiring a

Semiring

Instances

Semiring Bool

Semiring Double

Semiring Float

Semiring Int

Semiring Integer

(AdditiveGroup a, Semiring a) => Semiring (Complex a)

(KnownNat n, Semiring a) => Semiring (Vector n a)

(KnownNat n, Semiring a) => Semiring (Matrix Nat Nat n n a)

class (AdditiveGroup a, MultiplicativeAssociative a, MultiplicativeUnital a, Distribution a) => Ring a

Ring a summary of the laws inherited from the ring super-classes

zero + a == a
a + zero == a
(a + b) + c == a + (b + c)
a + b == b + a
a - a = zero
negate a = zero - a
negate a + a = zero
a + negate a = zero
one `times` a == a
a `times` one == a
(a `times` b) `times` c == a `times` (b `times` c)
a `times` (b + c) == a `times` b + a `times` c
(a + b) `times` c == a `times` c + b `times` c
a `times` zero == zero
zero `times` a == zero

Instances

Ring Double

Ring Float

Ring Int

Ring Integer

Ring a => Ring (Complex a)

(KnownNat n, Ring a) => Ring (Vector n a)

(KnownNat n, AdditiveGroup a, Semiring a) => Ring (Matrix Nat Nat n n a)

class (Multiplicative a, Ring a) => CRing a

CRing is a Ring with Multiplicative Commutation. It arises often due to * being defined as a multiplicative commutative operation.

Instances

CRing Double

CRing Float

CRing Int

CRing Integer

CRing a => CRing (Complex a)

(KnownNat n, CRing a) => CRing (Vector n a)