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  @niles -- Thanks again for your efforts. I filed the ticked (#10949).
  Spec means the same thing in Sage that it does everywhere else. It's just that
  Spec just doesn't check to see whether its input is a prime ideal: it relies on SchemeTopologicalPoint_prime_ideal: You can tell this because the __call__ method of sage.schemes.generic.spec.Spec is simply
  Spec
  SchemeTopologicalPoint_prime_ideal
  __call__
  sage.schemes.generic.spec.Spec
  return point.SchemeTopologicalPoint_prime_ideal(self, x)
  However SchemeTopologicalPoint_prime_ideal doesn't check to see whether the input ideal is prime either! It does allow an optional argument check which will perform the check, but this is disabled by default. Here is the code from sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal.__init__:
  check
  sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal.__init__
  R = S.coordinate_ring()
  from sage.rings.ideal import Ideal
  P = Ideal(R, P)
  # ideally we would have check=True by default, but
  # unfortunately is_prime() is only implemented in a small
  # number of cases
  if check and not P.is_prime():
  raise ValueError, "The argument %s must be a prime ideal of %s"%(P, R)
  SchemeTopologicalPoint.__init__(self, S)
  self.__P = P
  So if you were calling SchemeTopologicalPoint_prime_ideal directly, you could pass check=True to have it check for you:
  check=True
  sage: S = Spec(ZZ)
  sage: nZ = ZZ.ideal(6)
  sage: from sage.schemes.generic.point import SchemeTopologicalPoint_prime_ideal as primept
  sage: primept(S,nZ)
  Point on Spectrum of Integer Ring defined by the Principal ideal (6) of Integer Ring
  sage: primept(S,nZ,check=True)
  ---------------------------------------------------------------------------
  ValueError Traceback (most recent call last)
  ...
  ValueError: The argument Principal ideal (6) of Integer Ring must be a prime ideal of Integer Ring
  Unfortunately, the __call__ method of Spec doesn't take a check argument, and it doesn't pass its additional keyword arguments along using **kwds, so there isn't a way to have Spec check for you directly.
  **kwds
  To me, this all seems confusing, shoddy, and disappointing; you should file a ticket on Trac for this (if there isn't one already). If you're in a situation where you need this functionality, I would suggest adding a line of code to check whether the ideal is prime before you pass it to Spec.
  @John Palmieri. I shouldn't fill a trac ticket then? Thanks for your efforts.
  @niles It makes perfect sense now. Thank you.
  The
  i4: nZ.is_prime()
  o4: False
  Obviously, nZ is not a prime ideal, as 6 is composite. Hence by definition, it is not in $Spec(\mathbb{Z})$. So what does "Point on Spectrum" means exactly in Sage?
  Thanks.