Numeric.Ring.Rng

Documentation

data RngRing r

The free Ring given a Rng obtained by adjoining Z, such that

RngRing r = n*1 + r

This ring is commonly denoted r^.

Constructors

RngRing !Integer r

Instances

(Abelian r, Group r) => RightModule Integer (RngRing r)

Methods

(*.) :: RngRing r -> Integer -> RngRing r

(Abelian r, Monoidal r) => RightModule Natural (RngRing r)

Methods

(*.) :: RngRing r -> Natural -> RngRing r

(Abelian r, Group r) => LeftModule Integer (RngRing r)

Methods

(.*) :: Integer -> RngRing r -> RngRing r

(Abelian r, Monoidal r) => LeftModule Natural (RngRing r)

Methods

(.*) :: Natural -> RngRing r -> RngRing r

Read r => Read (RngRing r)

Methods

readsPrec :: Int -> ReadS (RngRing r)

readList :: ReadS [RngRing r]

readPrec :: ReadPrec (RngRing r)

readListPrec :: ReadPrec [RngRing r]

Show r => Show (RngRing r)

Methods

showsPrec :: Int -> RngRing r -> ShowS

show :: RngRing r -> String

showList :: [RngRing r] -> ShowS

Abelian r => Abelian (RngRing r)

Abelian r => Additive (RngRing r)

Methods

(+) :: RngRing r -> RngRing r -> RngRing r

sinnum1p :: Natural -> RngRing r -> RngRing r

sumWith1 :: Foldable1 f => (a -> RngRing r) -> f a -> RngRing r

(Abelian r, Monoidal r) => Monoidal (RngRing r)

Methods

zero :: RngRing r

sinnum :: Natural -> RngRing r -> RngRing r

sumWith :: Foldable f => (a -> RngRing r) -> f a -> RngRing r

Rng r => Semiring (RngRing r)

Rng r => Multiplicative (RngRing r)

Methods

(*) :: RngRing r -> RngRing r -> RngRing r

pow1p :: RngRing r -> Natural -> RngRing r

productWith1 :: Foldable1 f => (a -> RngRing r) -> f a -> RngRing r

(Abelian r, Group r) => Group (RngRing r)

Methods

(-) :: RngRing r -> RngRing r -> RngRing r

negate :: RngRing r -> RngRing r

subtract :: RngRing r -> RngRing r -> RngRing r

times :: Integral n => n -> RngRing r -> RngRing r

Rng r => Unital (RngRing r)

Methods

one :: RngRing r

pow :: RngRing r -> Natural -> RngRing r

productWith :: Foldable f => (a -> RngRing r) -> f a -> RngRing r

(Rng r, Division r) => Division (RngRing r)

Methods

recip :: RngRing r -> RngRing r

(/) :: RngRing r -> RngRing r -> RngRing r

(\\) :: RngRing r -> RngRing r -> RngRing r

(^) :: Integral n => RngRing r -> n -> RngRing r

(Commutative r, Rng r) => Commutative (RngRing r)

Rng r => Rig (RngRing r)

Methods

fromNatural :: Natural -> RngRing r

Rng r => Ring (RngRing r)

Methods

fromInteger :: Integer -> RngRing r

Rng s => RightModule (RngRing s) (RngRing s)

Methods

(*.) :: RngRing s -> RngRing s -> RngRing s

Rng s => LeftModule (RngRing s) (RngRing s)

Methods

(.*) :: RngRing s -> RngRing s -> RngRing s

rngRingHom :: r -> RngRing r

The rng homomorphism from r to RngRing r

liftRngHom :: Ring s => (r -> s) -> RngRing r -> s

given a rng homomorphism from a rng r into a ring s, liftRngHom yields a ring homomorphism from the ring `r^` into s.