diff --git a/src/sage/matrix/matrix2.pyx b/src/sage/matrix/matrix2.pyx
index 7fd2108e4d2..3f022f421ee 100644
--- a/src/sage/matrix/matrix2.pyx
+++ b/src/sage/matrix/matrix2.pyx
@@ -105,6 +105,13 @@ from sage.matrix.matrix_misc import permanental_minor_polynomial
 # used to deprecate only adjoint method
 from sage.misc.superseded import deprecated_function_alias
 
+# temporary hack to silence the warnings from #34806
+from sage.rings.number_field.order import Order as NumberFieldOrder
+def ideal_or_fractional(R, *args):
+    if isinstance(R, NumberFieldOrder):
+        R = R.number_field()
+    return R.ideal(*args)
+
 _Fields = Fields()
 
 cdef class Matrix(Matrix1):
@@ -15993,7 +16000,7 @@ cdef class Matrix(Matrix1):
         try:
             for i in xrange(1, len(pivs)):
                 y = a[i][pivs[i]]
-                I = R.ideal(y)
+                I = ideal_or_fractional(R, y)
                 s = a[0][pivs[i]]
                 t = I.small_residue(s)
                 v = R( (s-t) / y)
@@ -17730,9 +17737,9 @@ def _smith_diag(d, transformation=True):
     else:
         left = right = None
     for i in xrange(n):
-        I = R.ideal(dp[i,i])
+        I = ideal_or_fractional(R, dp[i,i])
 
-        if I == R.unit_ideal():
+        if I == ideal_or_fractional(R, 1):
             if dp[i,i] != 1:
                 if transformation:
                     left.add_multiple_of_row(i,i,R(R(1)/(dp[i,i])) - 1)
@@ -17741,12 +17748,12 @@ def _smith_diag(d, transformation=True):
 
         for j in xrange(i+1,n):
             if dp[j,j] not in I:
-                t = R.ideal([dp[i,i], dp[j,j]]).gens_reduced()
+                t = ideal_or_fractional(R, [dp[i,i], dp[j,j]]).gens_reduced()
                 if len(t) > 1:
                     raise ArithmeticError
                 t = t[0]
                 # find lambda, mu such that lambda*d[i,i] + mu*d[j,j] = t
-                lamb = R(dp[i,i]/t).inverse_mod( R.ideal(dp[j,j]/t))
+                lamb = R(dp[i,i]/t).inverse_mod( ideal_or_fractional(R, dp[j,j]/t))
                 mu = R((t - lamb*dp[i,i]) / dp[j,j])
 
                 newlmat = dp.new_matrix(dp.nrows(), dp.nrows(), 1)
@@ -17821,18 +17828,15 @@ def _generic_clear_column(m):
     # [e,f]
     # is invertible over R
 
-    if a[0,0] != 0:
-        I = R.ideal(a[0, 0]) # need to make sure we change this when a[0,0] changes
-    else:
-        I = R.zero_ideal()
+    I = ideal_or_fractional(R, a[0, 0]) # need to make sure we change this when a[0,0] changes
     for k in xrange(1, a.nrows()):
         if a[k,0] not in I:
             try:
-                v = R.ideal(a[0,0], a[k,0]).gens_reduced()
+                v = ideal_or_fractional(R, a[0,0], a[k,0]).gens_reduced()
             except Exception as msg:
                 raise ArithmeticError("%s\nCan't create ideal on %s and %s" % (msg, a[0,0], a[k,0]))
             if len(v) > 1:
-                raise ArithmeticError("Ideal %s not principal" % R.ideal(a[0,0], a[k,0]))
+                raise ArithmeticError("Ideal %s not principal" % ideal_or_fractional(R, a[0,0], a[k,0]))
             B = v[0]
 
             # now we find c,d, using the fact that c * (a_{0,0}/B) - d *
@@ -17841,7 +17845,7 @@ def _generic_clear_column(m):
             # need to handle carefully the case when a_{k,0}/B is a unit, i.e. a_{k,0} divides
             # a_{0,0}.
 
-            c = R(a[0,0] / B).inverse_mod(R.ideal(a[k,0] / B))
+            c = R(a[0,0] / B).inverse_mod(ideal_or_fractional(R, a[k,0] / B))
             d = R( (c*a[0,0] - B)/(a[k,0]) )
 
             # sanity check
@@ -17850,7 +17854,7 @@ def _generic_clear_column(m):
 
             # now we find e,f such that e*d + c*f = 1 in the same way
             if c != 0:
-                e = d.inverse_mod( R.ideal(c) )
+                e = d.inverse_mod( ideal_or_fractional(R, c) )
                 f = R((1 - d*e)/c)
             else:
                 e = R(-a[k,0]/B) # here d is a unit and this is just 1/d
@@ -17866,7 +17870,7 @@ def _generic_clear_column(m):
             if newlmat.det() != 1:
                 raise ArithmeticError
             a = newlmat*a
-            I = R.ideal(a[0,0])
+            I = ideal_or_fractional(R, a[0,0])
             left_mat = newlmat*left_mat
             if left_mat * m != a:
                 raise ArithmeticError
diff --git a/src/sage/modular/dirichlet.py b/src/sage/modular/dirichlet.py
index 7efa9d8553f..47f116abdd6 100644
--- a/src/sage/modular/dirichlet.py
+++ b/src/sage/modular/dirichlet.py
@@ -2339,6 +2339,7 @@ class DirichletGroupFactory(UniqueFactory):
         sage: parent(val)
         Gaussian Integers in Cyclotomic Field of order 4 and degree 2
         sage: r4.residue_field(r4.ideal(29).factor()[0][0])(val)
+        doctest:warning ... DeprecationWarning: ...
         17
         sage: r4.residue_field(r4.ideal(29).factor()[0][0])(val) * GF(29)(3)
         22
diff --git a/src/sage/rings/number_field/bdd_height.py b/src/sage/rings/number_field/bdd_height.py
index b7c8c33d0be..d0f37a2b192 100644
--- a/src/sage/rings/number_field/bdd_height.py
+++ b/src/sage/rings/number_field/bdd_height.py
@@ -430,7 +430,6 @@ def bdd_height(K, height_bound, tolerance=1e-2, precision=53):
     if B < 1:
         return
     embeddings = K.places(prec=precision)
-    O_K = K.ring_of_integers()
     r1, r2 = K.signature()
     r = r1 + r2 - 1
     RF = RealField(precision)
@@ -486,7 +485,7 @@ def log_height_for_generators_approx(alpha, beta, Lambda):
         Return a lambda approximation h_K(alpha/beta)
         """
         delta = Lambda / (r + 2)
-        norm_log = delta_approximation(RR(O_K.ideal(alpha, beta).norm()).log(), delta)
+        norm_log = delta_approximation(RR(K.ideal(alpha, beta).norm()).log(), delta)
         log_ga = vector_delta_approximation(log_map(alpha), delta)
         log_gb = vector_delta_approximation(log_map(beta), delta)
         arch_sum = sum([max(log_ga[k], log_gb[k]) for k in range(r + 1)])
diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index 0d2271844ca..7678911988b 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -12534,7 +12534,7 @@ def map_Zmstar_to_Zm(h):
             p = u.lift()
             while not p.is_prime():
                 p += m
-            f = R.ideal(p).prime_factors()[0].residue_class_degree()
+            f = K.fractional_ideal(p).prime_factors()[0].residue_class_degree()
             h = g**f
             if h not in H:
                 Hgens += [h]
diff --git a/src/sage/rings/number_field/number_field_element.pyx b/src/sage/rings/number_field/number_field_element.pyx
index 09a51a1264a..1d296d46928 100644
--- a/src/sage/rings/number_field/number_field_element.pyx
+++ b/src/sage/rings/number_field/number_field_element.pyx
@@ -169,9 +169,8 @@ def _inverse_mod_generic(elt, I):
     """
     from sage.matrix.constructor import matrix
     R = elt.parent()
-    try:
-        I = R.ideal(I)
-    except ValueError:
+    I = R.number_field().fractional_ideal(I)
+    if not I.is_integral():
         raise ValueError("inverse is only defined modulo integral ideals")
     if I == 0:
         raise ValueError("inverse is not defined modulo the zero ideal")
@@ -1987,6 +1986,9 @@ cdef class NumberFieldElement(FieldElement):
             raise ArithmeticError("factorization of 0 is not defined")
 
         K = self.parent()
+        from .order import is_NumberFieldOrder
+        if is_NumberFieldOrder(K):
+            K = K.number_field()
         fac = K.ideal(self).factor()
         # Check whether all prime ideals in `fac` are principal
         for P,e in fac:
@@ -1999,6 +2001,29 @@ cdef class NumberFieldElement(FieldElement):
         from sage.structure.all import Factorization
         return Factorization(element_fac, unit=self/element_product)
 
+    def is_prime(self):
+        r"""
+        Test whether this number-field element is prime as
+        an algebraic integer.
+
+        Note that the behavior of this method differs from the behavior
+        of :meth:`~sage.structure.element.RingElement.is_prime` in a
+        general ring, according to which (number) fields would have no
+        nonzero prime elements.
+
+        EXAMPLES::
+
+            sage: K.<i> = NumberField(x^2+1)
+            sage: (1+i).is_prime()
+            True
+            sage: ((1+i)/2).is_prime()
+            False
+        """
+        if not self or not self.is_integral():
+            return False
+        I = self.number_field().fractional_ideal(self)
+        return I.is_prime()
+
     @coerce_binop
     def gcd(self, other):
         """
@@ -2068,7 +2093,8 @@ cdef class NumberFieldElement(FieldElement):
         if not is_NumberFieldOrder(R) or not R.is_maximal():
             raise NotImplementedError("gcd() for %r is not implemented" % R)
 
-        g = R.ideal(self, other).gens_reduced()
+        K = R.number_field()
+        g = K.fractional_ideal(self, other).gens_reduced()
         if len(g) > 1:
             raise ArithmeticError("ideal (%r, %r) is not principal, gcd is not defined" % (self, other) )
 
diff --git a/src/sage/rings/number_field/order.py b/src/sage/rings/number_field/order.py
index f0b0c0f9656..65c59a6256d 100644
--- a/src/sage/rings/number_field/order.py
+++ b/src/sage/rings/number_field/order.py
@@ -491,6 +491,14 @@ def ideal(self, *args, **kwds):
         """
         if not self.is_maximal():
             raise NotImplementedError("ideals of non-maximal orders not yet supported.")
+        from sage.misc.superseded import deprecation
+        deprecation(34806, 'In the future, constructing an ideal of the ring of '
+                           'integers of a number field will use an implementation '
+                           'compatible with ideals of other (non-maximal) orders, '
+                           'rather than returning an integral fractional ideal of '
+                           'its containing number field. Use .fractional_ideal(), '
+                           'together with an .is_integral() check if desired, to '
+                           'avoid your code breaking with future changes to Sage.')
         I = self.number_field().ideal(*args, **kwds)
         if not I.is_integral():
             raise ValueError("ideal must be integral; use fractional_ideal to create a non-integral ideal.")
diff --git a/src/sage/rings/padics/padic_valuation.py b/src/sage/rings/padics/padic_valuation.py
index 2d444d0e676..f773236f7fb 100644
--- a/src/sage/rings/padics/padic_valuation.py
+++ b/src/sage/rings/padics/padic_valuation.py
@@ -283,7 +283,7 @@ def create_key_and_extra_args_for_number_field_from_ideal(self, R, I, prime):
 
         EXAMPLES::
 
-            sage: GaussianIntegers().valuation(GaussianIntegers().ideal(2)) # indirect doctest
+            sage: GaussianIntegers().valuation(GaussianIntegers().number_field().fractional_ideal(2)) # indirect doctest
             2-adic valuation
 
         TESTS:
diff --git a/src/sage/rings/polynomial/polynomial_element.pyx b/src/sage/rings/polynomial/polynomial_element.pyx
index e8acc5a65b1..e7126c11791 100644
--- a/src/sage/rings/polynomial/polynomial_element.pyx
+++ b/src/sage/rings/polynomial/polynomial_element.pyx
@@ -135,7 +135,7 @@ from sage.categories.morphism cimport Morphism
 from sage.misc.superseded import deprecation_cython as deprecation, deprecated_function_alias
 from sage.misc.cachefunc import cached_method
 
-from sage.rings.number_field.order import is_NumberFieldOrder
+from sage.rings.number_field.order import is_NumberFieldOrder, Order as NumberFieldOrder
 from sage.categories.number_fields import NumberFields
 
 
@@ -5105,8 +5105,11 @@ cdef class Polynomial(CommutativeAlgebraElement):
             y = self._parent.quo(self).gen()
             from sage.groups.generic import order_from_multiple
             return n == order_from_multiple(y, n, n_prime_divs, operation="*")
+        elif isinstance(R, NumberFieldOrder):
+            K = R.number_field()
+            return K.fractional_ideal(self.coefficients()) == K.fractional_ideal(1)
         else:
-            return R.ideal(self.coefficients())==R.ideal(1)
+            return R.ideal(self.coefficients()) == R.ideal(1)
 
     def is_constant(self):
         """
diff --git a/src/sage/rings/quotient_ring.py b/src/sage/rings/quotient_ring.py
index 8f9bf928cc5..33df6cf2e37 100644
--- a/src/sage/rings/quotient_ring.py
+++ b/src/sage/rings/quotient_ring.py
@@ -290,7 +290,13 @@ def QuotientRing(R, I, names=None, **kwds):
         from sage.rings.polynomial.polynomial_ring_constructor import BooleanPolynomialRing_constructor as BooleanPolynomialRing
         kwds.pop('implementation')
         return BooleanPolynomialRing(R.ngens(), names=names, **kwds)
-    if not isinstance(I, ideal.Ideal_generic) or I.ring() != R:
+    # workaround to silence warning from #34806
+    from sage.rings.number_field.order import Order
+    if isinstance(R, Order):
+        if not R.is_maximal():
+            raise NotImplementedError('only implemented for maximal orders')
+        I = R.number_field().ideal(I)
+    elif not isinstance(I, ideal.Ideal_generic) or I.ring() != R:
         I = R.ideal(I)
     if I.is_zero():
         return R
@@ -444,7 +450,13 @@ def __init__(self, R, I, names, category=None):
         """
         if R not in _Rings:
             raise TypeError("The first argument must be a ring, but %s is not"%R)
-        if I not in R.ideal_monoid():
+        # workaround to silence warning from #34806
+        from sage.rings.number_field.order import Order
+        if isinstance(R, Order):
+            M = R.number_field().ideal_monoid()
+        else:
+            M = R.ideal_monoid()
+        if I not in M:
             raise TypeError("The second argument must be an ideal of the given ring, but %s is not"%I)
         self.__R = R
         self.__I = I
diff --git a/src/sage/schemes/elliptic_curves/ell_number_field.py b/src/sage/schemes/elliptic_curves/ell_number_field.py
index 3b31f9cf43f..cce47432b18 100644
--- a/src/sage/schemes/elliptic_curves/ell_number_field.py
+++ b/src/sage/schemes/elliptic_curves/ell_number_field.py
@@ -1499,10 +1499,11 @@ def conductor(self):
         # Note: for number fields other than QQ we could initialize
         # N=K.ideal(1) or N=OK.ideal(1), which are the same, but for
         # K == QQ it has to be ZZ.ideal(1).
-        OK = self.base_ring().ring_of_integers()
+        K = self.base_field()
+        N = ZZ.ideal(1) if K is QQ else K.fractional_ideal(1)
         self._conductor = prod([d.prime()**d.conductor_valuation()
                                 for d in self.local_data()],
-                               OK.ideal(1))
+                               N)
         return self._conductor
 
     def minimal_discriminant_ideal(self):
diff --git a/src/sage/schemes/elliptic_curves/heegner.py b/src/sage/schemes/elliptic_curves/heegner.py
index 298f44fedc6..54c2f5ca1f7 100644
--- a/src/sage/schemes/elliptic_curves/heegner.py
+++ b/src/sage/schemes/elliptic_curves/heegner.py
@@ -1657,7 +1657,7 @@ def ideal(self):
         (A,B,C) = f
         if A%c == 0:
             A, C = C, A
-        return K.maximal_order().ideal([A, (-B+c*sqrtD)/2])
+        return K.fractional_ideal([A, (-B+c*sqrtD)/2])
 
 ##     def __call__(self, z):
 ##         """
diff --git a/src/sage/schemes/projective/projective_point.py b/src/sage/schemes/projective/projective_point.py
index d94d108e5d6..36a21a78e46 100644
--- a/src/sage/schemes/projective/projective_point.py
+++ b/src/sage/schemes/projective/projective_point.py
@@ -36,7 +36,7 @@
 from sage.categories.number_fields import NumberFields
 _NumberFields = NumberFields()
 from sage.rings.fraction_field import FractionField
-from sage.rings.number_field.order import is_NumberFieldOrder
+from sage.rings.number_field.order import is_NumberFieldOrder, Order as NumberFieldOrder
 from sage.rings.qqbar import number_field_elements_from_algebraics
 from sage.rings.quotient_ring import QuotientRing_generic
 from sage.rings.rational_field import QQ
@@ -750,6 +750,8 @@ def global_height(self, prec=None):
                 raise TypeError("must be defined over an algebraic field")
             else:
                 K = P.codomain().base_ring()
+        if isinstance(K, NumberFieldOrder):
+            K = K.number_field()
         # first get rid of the denominators
         denom = lcm([xi.denominator() for xi in P])
         x = [xi * denom for xi in P]