units for rings in haskell in Num or Rational

Question

The Num class of haskell allows for quite general algebraic structures and looks like it's intended to be used to make rings. When speaking of a ring though, it's convenient to be able to explicitly mention its additional and multiplicative units - maybe Num.Zero and Num.One - is there such a thing to Num, another class that includes units or some other way that this is done?

Answer 1

If your instance of Num is a ring, one expects fromInteger to be a ring homomorphism and thus 0 and 1 will just work. This may not always hold. Num predates typeclasses having algebraic laws be the norm. Also, unfortunetly, many instances of Num are not rings (such as floating point numbers).

Num is not really a ring structure, since it also has "other stuff" like abs, signum and the (hopefully) ring homomorphism fromInteger. I tend to think of it as "probably ring-ish with some other stuff."

Example: the ring of Gaussian rationals

import Data.Ratio
import Data.Complex

type GaussianRational = Complex Rational

zero :: GaussianRational 
zero = 0

one :: GaussianRational
one = 1

EDIT: Since Z is initial in Ring, the idea of using fromInteger this way actually makes a lot of sense.

Answer 2

The entire algebra package is devoted to these kinds of purposes. For instance, we have

class (Rig r, Rng r) => Ring r

and the supporting cast

class (Semiring r, Unital r, Monoidal r) => Rig r
class (Group r, Semiring r) => Rng r
class Multiplicative r => Unital r
class (Additive r, Abelian r, Multiplicative r) => Semiring r
class (LeftModule Integer r, RightModule Integer r, Monoidal r) => Group r
class (LeftModule Natural m, RightModule Natural m) => Monoidal m
class (Semiring r, Additive m) => RightModule r m
class (Semiring r, Additive m) => LeftModule r m
class Multiplicative r
class Additive r
class Additive r => Abelian r

which is at least one way to build up a ring. If you're doing highly general algebra then algebra might be worth it, but most libraries just expect Num.