Spec().an_element() is wrong when the coordinate ring is not Z

[RalphieBoy]

The code (schemes/generic/scheme.py) does this:

  return self(self.coordinate_ring().zero_ideal())

if the scheme's coordinate ring is not the integers. The coordinate ring need not be a domain, in which case the zero ideal is not prime (hence not a point in the prime spectrum).

I am not clear on the proper fix. I suspect for random rings, it's not easy to cobble up a prime ideal. At a minimum, the above could be called when the ring is a domain.

Component: PLEASE CHANGE

Keywords: spec, coordinate ring

Issue created by migration from https://trac.sagemath.org/ticket/20338

[videlec]

comment:1

You might want to do

def _an_element_(self):
    from sage.categories.domains import Domains
    if self.coordinate_ring() in Domains():
          return self(self.coordinate_ring().zero_ideal())
    else:
        raise NotImplementedError

[RalphieBoy]

comment:2

Is an_element() always supposed to return "0" in some guise? In any case, if the ring is "known", we can handle it (PID, UFD, domain, ...), no?

[videlec]

comment:3

Replying to @RalphieBoy:

Is an_element() always supposed to return "0" in some guise?

Not at all. This method is intended to be used in testing suites (i.e. when you perform TestSuite(my_object).run()). The less trivial the answer is, the best it is.

In any case, if the ring is "known", we can handle it (PID, UFD, domain, ...), no?

Sure. I was just proposing the laziest solution. You might even want to implement a method some_elements that is also used in the test suites.