-- | Finitely generated ideals in commutative rings.
module Algebra.Ideal
( Ideal(Id)
, zeroIdeal, isPrincipal, fromId
, eval, addId, mulId
, isSameIdeal, zeroIdealWitnesses
) where
import Data.List (intersperse,nub)
import Test.QuickCheck
import Algebra.Structures.CommutativeRing
-------------------------------------------------------------------------------
-- | Ideals characterized by their list of generators.
data CommutativeRing a => Ideal a = Id [a]
instance (CommutativeRing a, Show a) => Show (Ideal a) where
show (Id xs) = "<" ++ concat (intersperse "," (map show xs)) ++ ">"
instance (CommutativeRing a, Arbitrary a, Eq a) => Arbitrary (Ideal a) where
arbitrary = do xs' <- arbitrary
let xs = filter (/= zero) xs'
if xs == [] then return (Id [one]) else return (Id (nub xs))
-- | The zero ideal.
zeroIdeal :: CommutativeRing a => Ideal a
zeroIdeal = Id [zero]
-- | Test if an ideal is principal.
isPrincipal :: CommutativeRing a => Ideal a -> Bool
isPrincipal (Id xs) = length xs == 1
fromId :: CommutativeRing a => Ideal a -> [a]
fromId (Id xs) = xs
-- | Evaluate the ideal at a certain point.
eval :: CommutativeRing a => a -> Ideal a -> a
eval x (Id xs) = foldr (<+>) zero (map (<*> x) xs)
-- | Addition of ideals.
addId :: (CommutativeRing a, Eq a) => Ideal a -> Ideal a -> Ideal a
addId (Id xs) (Id ys) = Id (nub (xs ++ ys))
-- | Multiplication of ideals.
mulId :: (CommutativeRing a, Eq a) => Ideal a -> Ideal a -> Ideal a
mulId (Id xs) (Id ys) = if zs == [] then zeroIdeal else Id zs
where zs = nub [ f <*> g | f <- xs, g <- ys, f <*> g /= zero ]
{- | Test if an operations compute the correct ideal.
The operation should give a witness that the comuted ideal contains
the same elements.
If \[ x_1, ..., x_n \] \`op\` \[ y_1, ..., y_m \] = \[ z_1, ..., z_l \]
Then the witness should give that
z_k = a_k1 * x_1 + ... + a_kn * x_n
= b_k1 * y_1 + ... + b_km * y_m
This is used to check that the intersection computed is correct.
-}
isSameIdeal :: (CommutativeRing a, Eq a)
=> (Ideal a -> Ideal a -> (Ideal a, [[a]], [[a]]))
-> Ideal a
-> Ideal a
-> Bool
isSameIdeal op (Id xs) (Id ys) =
let (Id zs, as, bs) = (Id xs) `op` (Id ys)
in length as == length zs && length bs == length zs
&&
and [ z_k == sumRing (zipWith (<*>) a_k xs) && length a_k == length xs
| (z_k,a_k) <- zip zs as ]
&&
and [ z_k == sumRing (zipWith (<*>) b_k ys) && length b_k == length ys
| (z_k,b_k) <- zip zs bs ]
-- | Compute witnesses for two lists for the zero ideal. This is used when
-- computing the intersection of two ideals.
zeroIdealWitnesses :: (CommutativeRing a) => [a] -> [a] -> (Ideal a, [[a]], [[a]])
zeroIdealWitnesses xs ys = ( zeroIdeal
, [replicate (length xs) zero]
, [replicate (length ys) zero])