Safe Haskell	None
Language	Haskell2010

Algebra.Ring.Euclidean.Quotient

Description

Quotient rings for eulidean domains.

See Algebra.Ring.Polynomial.Quotient in halg-polynomial for an ideal quotient rings of polynomial ring.

Documentation

data Quotient r a Source #

Instances

Instances details

(Euclidean a, RightModule c a, Reifies r a) => RightModule c (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods (*.) :: Quotient r a -> c -> Quotient r a #
(Euclidean a, LeftModule c a, Reifies r a) => LeftModule c (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods (.*) :: c -> Quotient r a -> Quotient r a #
Eq a => Eq (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods (==) :: Quotient r a -> Quotient r a -> Bool # (/=) :: Quotient r a -> Quotient r a -> Bool #
Ord a => Ord (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods compare :: Quotient r a -> Quotient r a -> Ordering # (<) :: Quotient r a -> Quotient r a -> Bool # (<=) :: Quotient r a -> Quotient r a -> Bool # (>) :: Quotient r a -> Quotient r a -> Bool # (>=) :: Quotient r a -> Quotient r a -> Bool # max :: Quotient r a -> Quotient r a -> Quotient r a # min :: Quotient r a -> Quotient r a -> Quotient r a #
(Show a, Reifies r a) => Show (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods showsPrec :: Int -> Quotient r a -> ShowS # show :: Quotient r a -> String # showList :: [Quotient r a] -> ShowS #
(Euclidean a, Reifies r a) => Ring (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods fromInteger :: Integer -> Quotient r a
(Euclidean a, Reifies r a) => Rig (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods fromNatural :: Natural -> Quotient r a #
(Euclidean a, Reifies r a) => DecidableZero (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods isZero :: Quotient r a -> Bool #
(Euclidean a, Reifies r a) => DecidableUnits (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods recipUnit :: Quotient r a -> Maybe (Quotient r a) # isUnit :: Quotient r a -> Bool # (^?) :: Integral n => Quotient r a -> n -> Maybe (Quotient r a) #
(Euclidean a, Reifies r a) => DecidableAssociates (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods isAssociate :: Quotient r a -> Quotient r a -> Bool #
(Euclidean a, Reifies r a) => Unital (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods one :: Quotient r a # pow :: Quotient r a -> Natural -> Quotient r a # productWith :: Foldable f => (a0 -> Quotient r a) -> f a0 -> Quotient r a #
(Euclidean a, Reifies r a) => Commutative (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient
(Euclidean a, Reifies r a) => Semiring (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient
(Euclidean a, Reifies r a) => Multiplicative (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods (*) :: Quotient r a -> Quotient r a -> Quotient r a # pow1p :: Quotient r a -> Natural -> Quotient r a # productWith1 :: Foldable1 f => (a0 -> Quotient r a) -> f a0 -> Quotient r a #
(Euclidean a, Reifies r a) => Monoidal (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods zero :: Quotient r a # sinnum :: Natural -> Quotient r a -> Quotient r a # sumWith :: Foldable f => (a0 -> Quotient r a) -> f a0 -> Quotient r a #
(Euclidean a, Reifies r a) => Group (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods (-) :: Quotient r a -> Quotient r a -> Quotient r a # negate :: Quotient r a -> Quotient r a # subtract :: Quotient r a -> Quotient r a -> Quotient r a # times :: Integral n => n -> Quotient r a -> Quotient r a #
(Euclidean a, Reifies r a) => Additive (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods (+) :: Quotient r a -> Quotient r a -> Quotient r a # sinnum1p :: Natural -> Quotient r a -> Quotient r a # sumWith1 :: Foldable1 f => (a0 -> Quotient r a) -> f a0 -> Quotient r a #
(Euclidean a, Reifies r a) => Abelian (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient
(Show a, Reifies r a, DecidableZero a) => PrettyCoeff (Quotient r a) Source #
Instance details Defined in Algebra.Ring.Euclidean.Quotient Methods showsCoeff :: Int -> Quotient r a -> ShowSCoeff Source #

representative :: Quotient r a -> a Source #

withQuotient :: forall a. Euclidean a => a -> (forall (r :: Type). Reifies r a => Quotient r a) -> a Source #

reifyQuotient :: forall a r. Euclidean a => a -> (forall (s :: Type). Reifies s a => Proxy s -> r) -> r Source #

reifyIdealQuotient :: forall a r. Euclidean a => Ideal a -> (forall (s :: Type). Reifies s a => Proxy s -> r) -> r Source #

withIdealQuotient :: Euclidean a => Ideal a -> (forall (r :: Type). Reifies r a => Quotient r a) -> a Source #

quotient :: forall r a. (Reifies r a, Euclidean a) => a -> Quotient r a Source #

quotientBy :: forall r a proxy. (Reifies r a, Euclidean a) => proxy r -> a -> Quotient r a Source #