Incorrect blanket trait implementations for commutative rings

[grover-lc]

Consider the following program:

use derive_more::{Add, AddAssign, Mul, MulAssign, Sub, SubAssign};
use noether::{
    AssociativeAddition, AssociativeMultiplication, CommutativeAddition, CommutativeMultiplication,
    Distributive, PrincipalIdealDomain,
};
use num_traits::{One, Zero};
use std::ops::Neg;

// create and implement the ring ZZ x ZZ

#[derive(Debug, PartialEq, Add, AddAssign, Sub, SubAssign, Mul, MulAssign)]
#[mul(forward)]
#[mul_assign(forward)]
struct ZxZ(isize, isize);

impl AssociativeAddition for ZxZ {}
impl CommutativeAddition for ZxZ {}
impl AssociativeMultiplication for ZxZ {}
impl CommutativeMultiplication for ZxZ {}
impl Distributive for ZxZ {}
impl One for ZxZ {
    fn one() -> Self {
        ZxZ(1, 1)
    }
}
impl Zero for ZxZ {
    fn zero() -> Self {
        ZxZ(0, 0)
    }
    fn is_zero(&self) -> bool {
        self == &Self::zero()
    }
}
impl Neg for ZxZ {
    type Output = Self;
    fn neg(self) -> Self::Output {
        ZxZ(-self.0, -self.1)
    }
}

// a function which only accepts an input of a type implementing PrincipalIdealDomain
fn is_pid<T: PrincipalIdealDomain>(_: T) {
    println!("My type implements PrincipalIdealDomain.")
}

// mainline shows that ZZ x ZZ implements PrincipalIdealDomain
fn main() {
    let pair = ZxZ(5, -3);
    is_pid(pair);
}

Behaviour

This program compiles and runs without error, printing My type implements PrincipalIdealDomain..

Expected behaviour

I think the code should fail to compile because the explicitly implemented traits should not cause PrincipalIdealDomain to be implemented for ZxZ.

Note

The struct ZxZ represents the commutative ring $\mathbb{Z}\times\mathbb{Z}$. Only appropriate traits for this ring were explicitly implemented in this program, but $\mathbb{Z}\times\mathbb{Z}$ is not a principal ideal domain. It is not even an integral domain.

Proposal

Remove the following blanket implementations of IntegralDomain, UniqueFactorizationDomain and PrincipalIdealDomain for all types implementing CommutativeRing:

noether/src/rings.rs

Lines 107 to 109 in 677edcb

	impl<T: CommutativeRing> IntegralDomain for T {}
	impl<T: IntegralDomain> UniqueFactorizationDomain for T {}
	impl<T: UniqueFactorizationDomain> PrincipalIdealDomain for T {}