Algebra.Structures.Ring

The representation of the ring structure.

Documentation

class Ring a where

Definition of rings.

Methods

(<+>) :: a -> a -> a

Addition

(<*>) :: a -> a -> a

Multiplication

neg :: a -> a

Compute additive inverse

zero :: a

The additive identity

one :: a

The multiplicative identity

Instances

Ring Z

Ring EllipticCurve

Ring ZSqrt5

(GCDDomain a, Eq a) => Ring (FieldOfFractions a)

Nat n => Ring (Zn n)

(CommutativeRing r, Eq r) => Ring (UPoly r x)

propAddAssoc :: (Ring a, Eq a) => a -> a -> a -> (Bool, String)

Addition is associative.

propAddIdentity :: (Ring a, Eq a) => a -> (Bool, String)

Zero is the additive identity.

propAddInv :: (Ring a, Eq a) => a -> (Bool, String)

Negation give the additive inverse.

propAddComm :: (Ring a, Eq a) => a -> a -> (Bool, String)

Addition is commutative.

propMulAssoc :: (Ring a, Eq a) => a -> a -> a -> (Bool, String)

Multiplication is associative.

propMulIdentity :: (Ring a, Eq a) => a -> (Bool, String)

One is the multiplicative identity.

propRightDist :: (Ring a, Eq a) => a -> a -> a -> (Bool, String)

Multiplication is right-distributive over addition.

propLeftDist :: (Ring a, Eq a) => a -> a -> a -> (Bool, String)

Multiplication is left-ditributive over addition.

propRing :: (Ring a, Eq a) => a -> a -> a -> Property

Specification of rings. Test that the arguments satisfy the ring axioms.

(<->) :: Ring a => a -> a -> a

Subtraction

(<^>) :: Ring a => a -> Integer -> a

Exponentiation

sumRing :: Ring a => [a] -> a

Summation

productRing :: Ring a => [a] -> a

Product