Ring not recognized as a PID
I need to work with finitely generated modules over a PID, which the localization at a prime of the ring of Gaussian integers. It happens to be a PID, but if I try to execute the following code:
K.<i> = NumberField(x^2 + 1)
R = K.maximal_order().localization(7)
MS = span([[0, 1, 0]], R)
I get this error:
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-41-97a6c80cace8> in <module>
----> 1 MS = span([[Integer(0),Integer(1),Integer(0)]],R)
/opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/modules/free_module.py in span(gens, base_ring, check, already_echelonized)
686 if R not in PrincipalIdealDomains():
687 raise TypeError("The base_ring (= %s) must be a principal ideal "
--> 688 "domain." % R)
689 if not gens:
690 return FreeModule(R, 0)
TypeError: The base_ring (= Gaussian Integers in Number Field in i with defining polynomial x^2 + 1 localized at (7,)) must be a principal ideal domain.
I have tried with R = K.maximal_order()
and R = ZZ.localization(5)
, but they yield similar errors. So I understand that Sage doesn't automatically recognize those rings as PIDs, even though they are and there is a constructive way to find the generator of an ideal.
Is there a way for me to get Sage to see my ring as a PID?
Note that Sage does not even recognize the maximal order as a PID
There would be a lot to do around orders in number fields in Sage.