algebra-4.3: Constructive abstract algebra

Numeric.Ring.Rng

Synopsis

Documentation

data RngRing r Source #

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) Source # Methods(*.) :: RngRing r -> Integer -> RngRing r Source # (Abelian r, Monoidal r) => RightModule Natural (RngRing r) Source # Methods(*.) :: RngRing r -> Natural -> RngRing r Source # (Abelian r, Group r) => LeftModule Integer (RngRing r) Source # Methods(.*) :: Integer -> RngRing r -> RngRing r Source # (Abelian r, Monoidal r) => LeftModule Natural (RngRing r) Source # Methods(.*) :: Natural -> RngRing r -> RngRing r Source # Read r => Read (RngRing r) Source # MethodsreadsPrec :: Int -> ReadS (RngRing r) # Show r => Show (RngRing r) Source # MethodsshowsPrec :: Int -> RngRing r -> ShowS #show :: RngRing r -> String #showList :: [RngRing r] -> ShowS # Abelian r => Abelian (RngRing r) Source # Abelian r => Additive (RngRing r) Source # Methods(+) :: RngRing r -> RngRing r -> RngRing r Source #sinnum1p :: Natural -> RngRing r -> RngRing r Source #sumWith1 :: Foldable1 f => (a -> RngRing r) -> f a -> RngRing r Source # (Abelian r, Monoidal r) => Monoidal (RngRing r) Source # Methodssinnum :: Natural -> RngRing r -> RngRing r Source #sumWith :: Foldable f => (a -> RngRing r) -> f a -> RngRing r Source # Rng r => Semiring (RngRing r) Source # Rng r => Multiplicative (RngRing r) Source # Methods(*) :: RngRing r -> RngRing r -> RngRing r Source #pow1p :: RngRing r -> Natural -> RngRing r Source #productWith1 :: Foldable1 f => (a -> RngRing r) -> f a -> RngRing r Source # (Abelian r, Group r) => Group (RngRing r) Source # Methods(-) :: RngRing r -> RngRing r -> RngRing r Source #negate :: RngRing r -> RngRing r Source #subtract :: RngRing r -> RngRing r -> RngRing r Source #times :: Integral n => n -> RngRing r -> RngRing r Source # Rng r => Unital (RngRing r) Source # Methodspow :: RngRing r -> Natural -> RngRing r Source #productWith :: Foldable f => (a -> RngRing r) -> f a -> RngRing r Source # (Rng r, Division r) => Division (RngRing r) Source # Methodsrecip :: RngRing r -> RngRing r Source #(/) :: RngRing r -> RngRing r -> RngRing r Source #(\\) :: RngRing r -> RngRing r -> RngRing r Source #(^) :: Integral n => RngRing r -> n -> RngRing r Source # Rng r => Rig (RngRing r) Source # Methods Rng r => Ring (RngRing r) Source # Methods (Commutative r, Rng r) => Commutative (RngRing r) Source # Rng s => RightModule (RngRing s) (RngRing s) Source # Methods(*.) :: RngRing s -> RngRing s -> RngRing s Source # Rng s => LeftModule (RngRing s) (RngRing s) Source # Methods(.*) :: RngRing s -> RngRing s -> RngRing s Source #

rngRingHom :: r -> RngRing r Source #

The rng homomorphism from r to RngRing r

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

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