Add conformance tests for is_unit and is_nilpotent

[fingolfin]

Some ideas (assuming we exclude zero rings at the start):

@test is_unit(one(R))
@test !is_unit(zero(R))
@test is_nilpotent(zero(R))
@test !is_nilpotent(one(R))

For a random element x:

@test !(is_unit(x) && is_nilpotent(x))
@test is_unit(x) == is_unit(x^2)
@test is_nilpotent(x) == is_nilpotent(x^2)
if characteristic(R) > 0 && is_nilpotent(x)
  @test is_unit(x+1)
end
if is_domain_type(typeof(x))
  @test is_nilpotent(x) == is_zero(x)
end

Then we have of course tests of polynomials rings over a ring -- those could also have some tests for specific polynomials, i.e. if the indeterminate is t then is_unit(1+t) == false, etc.

[JohnAAbbott]

I propose testing also minus one:

@test is_unit(-one(R))
@test !is_nilpotent(-one(R))

And perhaps also along similar lines:

@test is_unit(x) == is_unit(-x)
@test is_nilpotent(x) == is_nilpotent(-x)

[JohnAAbbott]

After talking to @fingolfin here is a simple function for generating a nilpotent element: it can be very slow (since it factorizes) and works only if the modulus is a single value.

function nilpotent(R::ResidueRing{T})  where {T}
  facs = factor(modulus(R))  # might take a long time!
  rad = prod([fac_pow[1]  for fac_pow in facs])  # radical of the modulus
  return R(rad)  # zero if modulus is square-free
end

Bother: it does not work for ZZModRing because that is not a ResidueRing. Must we duplicate the code?

[fingolfin]

Regarding ZZModRing not being ResidueRing, see Nemocas/Nemo.jl#1819 (there are annoying technical issues with regards to rand which stalled this).

Regarding generating nilpotent elements and units for testing purposes: my idea was to have a pair of methods complementing test_elem, like so (untested code, just to demonstrate the idea):

function test_elem_nilpotent(R::Ring)
  return zero(R)
end

function test_elem_nilpotent(R::ResidueRing)
  error("TODO: factor modulus")
end

function test_elem_nilpotent(R::PolyRing)
  S = coefficient_ring(R)
  t = gen(R)
  return sum(test_elem_nilpotent(S)*t^i for i in 0:3)
end

function test_elem_nilpotent(R::MPolyRing)
  S = coefficient_ring(R)
  t = gen(R,1)
  # TODO: use all variables / generate random exponent vectors instead
  return sum(test_elem_nilpotent(S)*t^i for i in 0:3)
end

#
function test_elem_unit(R::Ring)
  if applicable(base_ring, R)
    S = base_ring(R)
    return R(test_elem_unit(S)) + test_elem_nilpotent(R)
  end
  return one(R) + test_elem_nilpotent(R)
end

BTW the function names are not great -- but I wanted something that riffs on test_elem. For better names I suggest we start by coming up with a better name for test_elem and then slowly migrate to that. But that's orthogonal to the core issue.

[fingolfin]

Once more regarding ZZModRing versus ResidueRing: yes for now we'll have to duplicate the code, but we'll survive this for one 5-line function :-). Maybe add one more line:

# TODO: get rid of this once ZZModRing derives from ResidueRing