diff --git a/src/sage/algebras/quatalg/quaternion_algebra.py b/src/sage/algebras/quatalg/quaternion_algebra.py
index 5a615b43978..34a6dfda820 100644
--- a/src/sage/algebras/quatalg/quaternion_algebra.py
+++ b/src/sage/algebras/quatalg/quaternion_algebra.py
@@ -2393,8 +2393,35 @@ 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):
         """
diff --git a/src/sage/modular/quatalg/brandt.py b/src/sage/modular/quatalg/brandt.py
index 1c74d953f6d..bfe7f71f8df 100644
--- a/src/sage/modular/quatalg/brandt.py
+++ b/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