gh-37113: Define behaviour of < and > for fractional ideals in a quat…

…ernion algebra

    
Previously, comparison of quaternion algebra fractional ideals (in a
quaternion algebra over QQ) was documented only for equality and did
compare the matrices under HNF of bases of the ideals. This gave strange
and undocumented behavior when comparing two ideals with inequalities.

This comparison is now replaced by the comparison of the fractional
ideals as free modules.

Now "smaller than" means "included in".

#sd123
    
URL: #37113
Reported by: syndrakon
Reviewer(s): David Coudert, Lorenz Panny

src/sage/algebras/quatalg/quaternion_algebra.py

@@ -2585,8 +2585,35 @@ def _richcmp_(self, right, op):
 
             sage: I != I                # indirect doctest
             False
+
+        TESTS::
+
+            sage: B = QuaternionAlgebra(QQ,-1,-11)
+            sage: i,j,k = B.gens()
+            sage: I = B.ideal([1,i,j,i*j])
+            sage: I == I
+            True
+            sage: O = B.ideal([1,i,(i+j)/2,(1+i*j)/2])
+            sage: I <= O
+            True
+            sage: I >= O
+            False
+            sage: I != O
+            True
+            sage: I == O
+            False
+            sage: I != I
+            False
+            sage: I < I
+            False
+            sage: I < O
+            True
+            sage: I <= I
+            True
+            sage: O >= O
+            True
         """
-        return self.basis_matrix()._richcmp_(right.basis_matrix(), op)
+        return self.free_module().__richcmp__(right.free_module(), op)
 
     def __hash__(self):
         """

src/sage/modular/quatalg/brandt.py

@@ -1365,7 +1365,10 @@ def right_ideals(self, B=None):
                             ideals_theta[J_theta] = [J]
                         verbose("found %s of %s ideals" % (len(ideals), self.dimension()), level=2)
                         if len(ideals) >= self.dimension():
-                            ideals = tuple(sorted(ideals))
+                            # order by basis matrix (as ideals were previously
+                            # ordered) for backward compatibility and
+                            # deterministic order of the output
+                            ideals = tuple(sorted(ideals, key=lambda x: x.basis_matrix()))
                             self.__right_ideals = ideals
                             return ideals
                         got_something_new = True