Add Ideal.radical and Ring.nilradical

src/sage/categories/rings.py

@@ -854,6 +854,21 @@ def zero_ideal(self):
             """
             return self._ideal_class_(1)(self, [self.zero()])
 
+        def nilradical(self):
+            """
+            Return the nilradical of this ring.
+
+            EXAMPLES::
+
+                sage: QQ['x,y'].nilradical()
+                Ideal (0) of Multivariate Polynomial Ring in x, y over Rational Field
+
+            .. SEEALSO::
+
+                :meth:`~sage.categories.finite_dimensional_lie_algebras_with_basis.ParentMethods.nilradical`
+            """
+            return self.zero_ideal().radical()
+
         @cached_method
         def unit_ideal(self):
             """

src/sage/rings/finite_rings/integer_mod_ring.py

@@ -750,12 +750,12 @@ def is_field(self, proof=None):
             self._factory_data[3]['category'] = Fields()
         else:
             if self.category().is_subcategory(Fields()):
-                raise ValueError("""THIS SAGE SESSION MIGHT BE SERIOUSLY COMPROMISED!
-The order {} is not prime, but this ring has been put
-into the category of fields. This may already have consequences
-in other parts of Sage. Either it was a mistake of the user,
-or a probabilistic primality test has failed.
-In the latter case, please inform the developers.""".format(self.order()))
+                raise ValueError(("THIS SAGE SESSION MIGHT BE SERIOUSLY COMPROMISED!\n"
+                    "The order {} is not prime, but this ring has been put\n"
+                    "into the category of fields. This may already have consequences\n"
+                    "in other parts of Sage. Either it was a mistake of the user,\n"
+                    "or a probabilistic primality test has failed.\n"
+                    "In the latter case, please inform the developers.").format(self.order()))
         return is_prime
 
     @cached_method

src/sage/rings/ideal.py

@@ -1695,6 +1695,17 @@ def residue_field(self):
             return ZZ.residue_field(self, check=False)
         raise NotImplementedError("residue_field() is only implemented for ZZ and rings of integers of number fields.")
 
+    def radical(self):
+        r"""
+        Return the radical of this ideal.
+
+        EXAMPLES::
+
+            sage: ZZ.ideal(12).radical()
+            Principal ideal (6) of Integer Ring
+        """
+        return self.ring().ideal(self.gen().radical())
+
 
 class Ideal_fractional(Ideal_generic):
     """

src/sage/rings/polynomial/multi_polynomial_ideal.py

@@ -244,6 +244,7 @@
 from sage.misc.misc_c import prod
 from sage.misc.verbose import get_verbose, verbose
 from sage.rings.ideal import Ideal_generic
+from sage.rings.quotient_ring import QuotientRingIdeal_generic
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.rings.noncommutative_ideals import Ideal_nc
@@ -5632,7 +5633,7 @@ def weil_restriction(self):
         return result_ring.ideal(result)
 
 
-class MPolynomialIdeal_quotient(MPolynomialIdeal):
+class MPolynomialIdeal_quotient(QuotientRingIdeal_generic, MPolynomialIdeal):
     r"""
     An ideal in a quotient of a multivariate polynomial ring.
 

src/sage/rings/quotient_ring.py

@@ -1505,6 +1505,27 @@
         Ideal (3, 0) of Ring of integers modulo 9
     """
 
+    def _lift(self):
+        """
+        Return an ideal of the cover ring that corresponds to this ideal.
+
+        EXAMPLES::
+
+            sage: Zmod(15).ideal(6)._lift()
+            Principal ideal (3) of Integer Ring
+            sage: R.<x,y> = QQ[]
+            sage: S = R.quotient(x)
+            sage: S.ideal(y)._lift()
+            Ideal (x, y) of Multivariate Polynomial Ring in x, y over Rational Field
+        """
+        R = self.ring()
+        if hasattr(R, 'defining_ideal'):
+            Igens = list(R.defining_ideal().gens())
+        else:
+            Igens = [R.modulus()]
+        Igens += [g.lift() for g in self.gens()]
+        return R.cover_ring().ideal(Igens)
+
     def _contains_(self, other):
         r"""
         Check whether this ideal contains the given element.
@@ -1534,15 +1555,19 @@
             sage: 5-5*t in S.ideal(t^2 - 1)
             True
         """
-        R = self.ring()
-        assert other in R
-        if hasattr(R, 'defining_ideal'):
-            Igens = list(R.defining_ideal().gens())
-        else:
-            Igens = [R.modulus()]
-        Igens += [g.lift() for g in self.gens()]
-        J = R.cover_ring().ideal(Igens)
-        return other.lift() in J
+        assert other in self.ring()
+        return other.lift() in self._lift()
+
+    def radical(self):
+        """
+        Return the radical of this ideal.
+
+        EXAMPLES::
+
+            sage: Zmod(16).ideal(4).radical()
+            Principal ideal (2) of Ring of integers modulo 16
+        """
+        return self.ring().ideal(self._lift().radical())
 
 
 class QuotientRingIdeal_principal(ideal.Ideal_principal, QuotientRingIdeal_generic):