is_trivial should be implemented for ideals in multivariate polynomial rings

[kedlaya]

This is easy to fix (by overriding the default method):

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: I = R.ideal(0,1)
sage: I == R.unit_ideal()
True
sage: I.is_trivial() # should give the same answer!
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-15-f115fda66859> in <module>()
----> 1 I.is_trivial()
 
/projects/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/sage/rings/ideal.pyc in is_trivial(self)
    945             return True
    946         # If self is principal, can give a complete answer
--> 947         if self.is_principal():
    948             return self.gens()[0].is_unit()
    949         # If self is not principal, can only give an affirmative answer
 
/projects/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/sage/rings/ideal.pyc in is_principal(self)
    906         if len(self.gens()) <= 1:
    907             return True
--> 908         raise NotImplementedError
    909
    910     def is_trivial(self):
 
NotImplementedError:

[kedlaya]

On second thought, would it be better to change the default method itself, to test equality against the unit_ideal of the parent ring? After all, if you are working with a ring where you can't test equality, then there's no sense in being able to test for triviality either.

[kedlaya]

The docstring for is_trivial states:

Return ``True`` if this ideal is `(0)` or `(1)`.

Setting aside the zero ring (where all possible interpretations seem to lead to the answer True), there seem to be three different definitions floating around for what constitutes a trivial ideal of a ring: the zero ideal; the unit ideal; or both. In commutative algebra, I'm generally used to seeing the second interpretation. The first interpretation is the one compatible with group theory, where triviality tends to refer exclusively to the one-element subgroup.

This ambiguity makes me wonder whether we should even provide this method at all! But if we do provide it, then it should be implemented using comparisons with the zero_ideal and/or unit_ideal of the parent ring.

[kedlaya]

This patch changes the implementation as proposed, without changing the mathematical definition. If there is interest in changing the mathematical definition, that should (be discussed elsewhere and then) happen on a separate ticket.

---

New commits:

​dce7e0b

Test is_trivial using comparison to unit ideal

[vdelecroix]

There must be a line break between the documentation string ...ticket :trac:`20514`:: and the begining of the tests.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​5e83ce7

Correct docstring formatting

[kedlaya]

Replying to vdelecroix:

There must be a line break between the documentation string ...ticket :trac:`20514`:: and the begining of the tests.

Corrected in the last commit.