diff --git a/src/sage/rings/asymptotic/asymptotic_ring.py b/src/sage/rings/asymptotic/asymptotic_ring.py
index 1c4158d92d4..fd25735f6a0 100644
--- a/src/sage/rings/asymptotic/asymptotic_ring.py
+++ b/src/sage/rings/asymptotic/asymptotic_ring.py
@@ -779,7 +779,7 @@ def __hash__(self):
         return hash(str(self))
 
 
-    def __nonzero__(self):
+    def __bool__(self):
         r"""
         Return whether this asymptotic expansion is not identically zero.
 
@@ -803,6 +803,7 @@ def __nonzero__(self):
         """
         return bool(self._summands_)
 
+    __nonzero__ =__bool__
 
     def __eq__(self, other):
         r"""
diff --git a/src/sage/rings/complex_arb.pyx b/src/sage/rings/complex_arb.pyx
index a43dbf63c97..6828eab8c16 100644
--- a/src/sage/rings/complex_arb.pyx
+++ b/src/sage/rings/complex_arb.pyx
@@ -1402,7 +1402,7 @@ cdef class ComplexBall(RingElement):
         return (arb_is_nonzero(acb_realref(self.value))
                 or arb_is_nonzero(acb_imagref(self.value)))
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` iff this complex ball is not the zero ball, i.e. if the
         midpoint and radius of its real and imaginary parts are not all zero.
@@ -1427,6 +1427,8 @@ cdef class ComplexBall(RingElement):
         """
         return not acb_is_zero(self.value)
 
+    __nonzero__ =__bool__
+
     def is_exact(self):
         """
         Return ``True`` iff the radius of this ball is zero.
diff --git a/src/sage/rings/complex_interval.pyx b/src/sage/rings/complex_interval.pyx
index 0a65cd58f2e..61828eef54d 100644
--- a/src/sage/rings/complex_interval.pyx
+++ b/src/sage/rings/complex_interval.pyx
@@ -1389,26 +1389,26 @@ cdef class ComplexIntervalFieldElement(sage.structure.element.FieldElement):
         """
         raise TypeError("can't convert complex interval to complex")
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` if ``self`` is not known to be exactly zero.
 
         EXAMPLES::
 
-            sage: CIF(RIF(0, 0), RIF(0, 0)).__nonzero__()
-            False
             sage: bool(CIF(RIF(0, 0), RIF(0, 0)))
             False
-            sage: CIF(RIF(1), RIF(0)).__nonzero__()
+            sage: bool(CIF(RIF(1), RIF(0)))
             True
-            sage: CIF(RIF(0), RIF(1)).__nonzero__()
+            sage: bool(CIF(RIF(0), RIF(1)))
             True
-            sage: CIF(RIF(1, 2), RIF(0)).__nonzero__()
+            sage: bool(CIF(RIF(1, 2), RIF(0)))
             True
-            sage: CIF(RIF(-1, 1), RIF(-1, 1)).__nonzero__()
+            sage: bool(CIF(RIF(-1, 1), RIF(-1, 1)))
             True
         """
-        return self.real().__nonzero__() or self.imag().__nonzero__()
+        return self.real().__bool__() or self.imag().__bool__()
+
+    __nonzero__ =__bool__
 
     cpdef _richcmp_(left, right, int op):
         r"""
diff --git a/src/sage/rings/complex_mpc.pyx b/src/sage/rings/complex_mpc.pyx
index 92e6ad9f700..aa9c88195a7 100644
--- a/src/sage/rings/complex_mpc.pyx
+++ b/src/sage/rings/complex_mpc.pyx
@@ -1243,7 +1243,7 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
             elif cim<0: return -1
             else: return 0
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` if ``self`` is not zero.
 
@@ -1258,6 +1258,8 @@ cdef class MPComplexNumber(sage.structure.element.FieldElement):
         """
         return not (mpfr_zero_p(self.value.re) and mpfr_zero_p(self.value.im))
 
+    __nonzero__ =__bool__
+
     def is_square(self):
         r"""
         This function always returns true as `\CC` is algebraically closed.
diff --git a/src/sage/rings/complex_number.pyx b/src/sage/rings/complex_number.pyx
index beadf16b075..cb68ef7e34d 100644
--- a/src/sage/rings/complex_number.pyx
+++ b/src/sage/rings/complex_number.pyx
@@ -859,7 +859,7 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
                                       real_string, digit_precision_bound,
                                       imag_string, digit_precision_bound)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` if ``self`` is not zero. This is an internal function;
         use :meth:`is_zero()` instead.
@@ -872,6 +872,8 @@ cdef class ComplexNumber(sage.structure.element.FieldElement):
         """
         return not (mpfr_zero_p(self.__re) and mpfr_zero_p(self.__im))
 
+    __nonzero__ =__bool__
+
     def prec(self):
         """
         Return precision of this complex number.
diff --git a/src/sage/rings/continued_fraction.py b/src/sage/rings/continued_fraction.py
index 4ce33c8f294..f3317a03188 100644
--- a/src/sage/rings/continued_fraction.py
+++ b/src/sage/rings/continued_fraction.py
@@ -986,7 +986,7 @@ def ceil(self):
             return self.quotient(0)
         return self.quotient(0)+1
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return False if self is zero.
 
@@ -1003,6 +1003,8 @@ def __nonzero__(self):
         """
         return bool(self.quotient(0)) or self.quotient(1) is not Infinity
 
+    __nonzero__ =__bool__
+
     def is_zero(self):
         r"""
         Test whether ``self`` is zero.
diff --git a/src/sage/rings/finite_rings/element_givaro.pyx b/src/sage/rings/finite_rings/element_givaro.pyx
index 7e077bc6e97..dafd3c8e624 100644
--- a/src/sage/rings/finite_rings/element_givaro.pyx
+++ b/src/sage/rings/finite_rings/element_givaro.pyx
@@ -926,7 +926,7 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
         """
         return self.element
 
-    def __nonzero__(FiniteField_givaroElement self):
+    def __bool__(FiniteField_givaroElement self):
         r"""
         Return ``True`` if ``self != k(0)``.
 
@@ -941,6 +941,8 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
         """
         return not self._cache.objectptr.isZero(self.element)
 
+    __nonzero__ =__bool__
+
     def is_one(FiniteField_givaroElement self):
         r"""
         Return ``True`` if ``self == k(1)``.
diff --git a/src/sage/rings/finite_rings/element_ntl_gf2e.pyx b/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
index b1ffd765c1e..19bad4bc1a6 100644
--- a/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
+++ b/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
@@ -574,7 +574,7 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
 
         return " + ".join(reversed(_repr))
 
-    def __nonzero__(FiniteField_ntl_gf2eElement self):
+    def __bool__(FiniteField_ntl_gf2eElement self):
         r"""
         Return ``True`` if ``self != k(0)``.
 
@@ -591,6 +591,8 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
         (<Cache_ntl_gf2e>self._parent._cache).F.restore()
         return not bool(GF2E_IsZero(self.x))
 
+    __nonzero__ =__bool__
+
     def is_one(FiniteField_ntl_gf2eElement self):
         r"""
         Return ``True`` if ``self == k(1)``.
diff --git a/src/sage/rings/finite_rings/element_pari_ffelt.pyx b/src/sage/rings/finite_rings/element_pari_ffelt.pyx
index a61d3e0b26d..55dc48f9b10 100644
--- a/src/sage/rings/finite_rings/element_pari_ffelt.pyx
+++ b/src/sage/rings/finite_rings/element_pari_ffelt.pyx
@@ -560,7 +560,8 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
         """
         return not bool(FF_equal0(self.val))
 
-    __nonzero__ = is_unit
+    __bool__ = is_unit
+    __nonzero__ =__bool__
 
     def __pos__(FiniteFieldElement_pari_ffelt self):
         """
diff --git a/src/sage/rings/finite_rings/integer_mod.pyx b/src/sage/rings/finite_rings/integer_mod.pyx
index 1b1d38852ae..6e6b479c109 100644
--- a/src/sage/rings/finite_rings/integer_mod.pyx
+++ b/src/sage/rings/finite_rings/integer_mod.pyx
@@ -1901,7 +1901,7 @@ cdef class IntegerMod_gmp(IntegerMod_abstract):
         """
         return mpz_cmp_si(self.value, 1) == 0
 
-    def __nonzero__(IntegerMod_gmp self):
+    def __bool__(IntegerMod_gmp self):
         """
         Returns ``True`` if this is not `0`, otherwise
         ``False``.
@@ -1915,6 +1915,8 @@ cdef class IntegerMod_gmp(IntegerMod_abstract):
         """
         return mpz_cmp_si(self.value, 0) != 0
 
+    __nonzero__ =__bool__
+
     cpdef bint is_unit(self):
         """
         Return True iff this element is a unit.
@@ -2297,7 +2299,7 @@ cdef class IntegerMod_int(IntegerMod_abstract):
         """
         return self.ivalue == 1
 
-    def __nonzero__(IntegerMod_int self):
+    def __bool__(IntegerMod_int self):
         """
         Returns ``True`` if this is not `0`, otherwise
         ``False``.
@@ -2311,6 +2313,8 @@ cdef class IntegerMod_int(IntegerMod_abstract):
         """
         return self.ivalue != 0
 
+    __nonzero__ =__bool__
+
     cpdef bint is_unit(IntegerMod_int self):
         """
         Return True iff this element is a unit
@@ -3114,7 +3118,7 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
         """
         return self.ivalue == 1
 
-    def __nonzero__(IntegerMod_int64 self):
+    def __bool__(IntegerMod_int64 self):
         """
         Returns ``True`` if this is not `0`, otherwise
         ``False``.
@@ -3128,6 +3132,8 @@ cdef class IntegerMod_int64(IntegerMod_abstract):
         """
         return self.ivalue != 0
 
+    __nonzero__ =__bool__
+
     cpdef bint is_unit(IntegerMod_int64 self):
         """
         Return True iff this element is a unit.
diff --git a/src/sage/rings/fraction_field_element.pyx b/src/sage/rings/fraction_field_element.pyx
index e2368fa0c3e..a8a0d395c63 100644
--- a/src/sage/rings/fraction_field_element.pyx
+++ b/src/sage/rings/fraction_field_element.pyx
@@ -869,7 +869,7 @@ cdef class FractionFieldElement(FieldElement):
         """
         return self.__numerator.valuation(v) - self.__denominator.valuation(v)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` if this element is nonzero.
 
@@ -878,16 +878,18 @@ cdef class FractionFieldElement(FieldElement):
             sage: F = ZZ['x,y'].fraction_field()
             sage: x,y = F.gens()
             sage: t = F(0)/x
-            sage: t.__nonzero__()
+            sage: bool(t)
             False
 
         ::
 
-            sage: (1/x).__nonzero__()
+            sage: bool(1/x)
             True
         """
         return not self.__numerator.is_zero()
 
+    __nonzero__ =__bool__
+
     def is_zero(self):
         """
         Return ``True`` if this element is equal to zero.
diff --git a/src/sage/rings/function_field/function_field_element.pyx b/src/sage/rings/function_field/function_field_element.pyx
index 21eb0ca74b7..b4c17de2df0 100644
--- a/src/sage/rings/function_field/function_field_element.pyx
+++ b/src/sage/rings/function_field/function_field_element.pyx
@@ -346,7 +346,7 @@ cdef class FunctionFieldElement_polymod(FunctionFieldElement):
         """
         return self._x._repr(name=self.parent().variable_name())
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         EXAMPLES::
 
@@ -361,6 +361,8 @@ cdef class FunctionFieldElement_polymod(FunctionFieldElement):
         """
         return not not self._x
 
+    __nonzero__ =__bool__
+
     def __hash__(self):
         """
         TESTS::
@@ -560,7 +562,7 @@ cdef class FunctionFieldElement_rational(FunctionFieldElement):
         """
         return repr(self._x)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         EXAMPLES::
 
diff --git a/src/sage/rings/ideal.py b/src/sage/rings/ideal.py
index a4e100b7c8e..e8815a1718a 100644
--- a/src/sage/rings/ideal.py
+++ b/src/sage/rings/ideal.py
@@ -386,7 +386,7 @@ def _contains_(self, x):
         """
         raise NotImplementedError
 
-    def __nonzero__(self):
+    def __bool__(self):
         r"""
         Return ``True`` if this ideal is not `(0)`.
 
@@ -419,6 +419,8 @@ def __nonzero__(self):
                 return True
         return False
 
+    __nonzero__ =__bool__
+
     def base_ring(self):
         r"""
         Returns the base ring of this ideal.
diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index d7369cf3d66..9fc26138ce2 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -4293,7 +4293,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         """
         return mpz_cmp_si(self.value, 1) == 0
 
-    def __nonzero__(self):
+    def __bool__(self):
         r"""
         Returns ``True`` if the integer is not `0`, otherwise ``False``.
 
@@ -4306,6 +4306,8 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         """
         return mpz_sgn(self.value) != 0
 
+    __nonzero__ =__bool__
+
     def is_integral(self):
         """
         Return ``True`` since integers are integral, i.e.,
diff --git a/src/sage/rings/laurent_series_ring_element.pyx b/src/sage/rings/laurent_series_ring_element.pyx
index 6c57351295e..e3ceded54cf 100644
--- a/src/sage/rings/laurent_series_ring_element.pyx
+++ b/src/sage/rings/laurent_series_ring_element.pyx
@@ -221,9 +221,11 @@ cdef class LaurentSeries(AlgebraElement):
 
         return self.__u.is_monomial()
 
-    def __nonzero__(self):
+    def __bool__(self):
         return not not self.__u
 
+    __nonzero__ =__bool__
+
     def _im_gens_(self, codomain, im_gens):
         return codomain(self(im_gens[0]))
 
diff --git a/src/sage/rings/morphism.pyx b/src/sage/rings/morphism.pyx
index 851408e69ad..7e25e0dfcfc 100644
--- a/src/sage/rings/morphism.pyx
+++ b/src/sage/rings/morphism.pyx
@@ -602,7 +602,7 @@ cdef class RingHomomorphism(RingMap):
             raise TypeError("parent must be a ring homset")
         RingMap.__init__(self, parent)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Every ring map is nonzero unless the domain or codomain is the
         0 ring, since there is no zero map between rings, since 1 goes
@@ -626,6 +626,8 @@ cdef class RingHomomorphism(RingMap):
         """
         return bool(self.codomain().one())
 
+    __nonzero__ =__bool__
+
     def _repr_type(self):
         """
         Used internally in printing this morphism.
diff --git a/src/sage/rings/multi_power_series_ring_element.py b/src/sage/rings/multi_power_series_ring_element.py
index 142f38b6d0a..4c4ef193f86 100644
--- a/src/sage/rings/multi_power_series_ring_element.py
+++ b/src/sage/rings/multi_power_series_ring_element.py
@@ -220,7 +220,7 @@ class MPowerSeries(PowerSeries):
     #
     # _mul_prec : works just fine
     #
-    # __nonzero__ : works just fine
+    # __bool__ : works just fine
     #
     """
     Multivariate power series; these are the elements of Multivariate Power
diff --git a/src/sage/rings/number_field/number_field_element.pyx b/src/sage/rings/number_field/number_field_element.pyx
index a03cce2d7c6..7bfad4cbf14 100644
--- a/src/sage/rings/number_field/number_field_element.pyx
+++ b/src/sage/rings/number_field/number_field_element.pyx
@@ -2265,27 +2265,27 @@ cdef class NumberFieldElement(FieldElement):
         """
         return self._mul_(~right)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return True if this number field element is nonzero.
 
         EXAMPLES::
 
             sage: m.<b> = CyclotomicField(17)
-            sage: m(0).__nonzero__()
+            sage: bool(m(0))
             False
-            sage: b.__nonzero__()
+            sage: bool(b)
             True
 
-        Nonzero is used by the bool command::
+        ``__bool__`` is used by the bool command::
 
             sage: bool(b + 1)
             True
-            sage: bool(m(0))
-            False
         """
         return not IsZero_ZZX(self.__numerator)
 
+    __nonzero__ =__bool__
+
     cpdef _neg_(self):
         r"""
         EXAMPLE::
diff --git a/src/sage/rings/number_field/number_field_element_quadratic.pyx b/src/sage/rings/number_field/number_field_element_quadratic.pyx
index efdcec9b6fc..0a5950ce158 100644
--- a/src/sage/rings/number_field/number_field_element_quadratic.pyx
+++ b/src/sage/rings/number_field/number_field_element_quadratic.pyx
@@ -1317,7 +1317,7 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         h += 42082631 * mpz_pythonhash(self.b)
         return h
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Check whether this element is not zero.
 
@@ -1331,6 +1331,8 @@ cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
         """
         return mpz_cmp_ui(self.a, 0) != 0 or mpz_cmp_ui(self.b, 0) != 0
 
+    __nonzero__ =__bool__
+
     def _integer_(self, Z=None):
         """
         EXAMPLES:
diff --git a/src/sage/rings/padics/CA_template.pxi b/src/sage/rings/padics/CA_template.pxi
index 5a23eb2e96e..c87b7fd9351 100644
--- a/src/sage/rings/padics/CA_template.pxi
+++ b/src/sage/rings/padics/CA_template.pxi
@@ -605,7 +605,7 @@ cdef class CAElement(pAdicTemplateElement):
                 return True
         return mpz_cmp_si((<Integer>absprec).value, val) <= 0
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Whether this element should be considered true in a boolean context.
 
@@ -621,6 +621,8 @@ cdef class CAElement(pAdicTemplateElement):
         """
         return not ciszero(self.value, self.prime_pow)
 
+    __nonzero__ =__bool__
+
     def is_equal_to(self, _right, absprec=None):
         r"""
         Determines whether the inputs are equal modulo
diff --git a/src/sage/rings/padics/CR_template.pxi b/src/sage/rings/padics/CR_template.pxi
index 2415d28e7d9..cb307c21ad3 100644
--- a/src/sage/rings/padics/CR_template.pxi
+++ b/src/sage/rings/padics/CR_template.pxi
@@ -959,7 +959,7 @@ cdef class CRElement(pAdicTemplateElement):
                 return True
         return mpz_cmp_si((<Integer>absprec).value, self.ordp) <= 0
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Returns True if self is distinguishable from zero.
 
@@ -975,6 +975,8 @@ cdef class CRElement(pAdicTemplateElement):
         """
         return self.relprec != 0
 
+    __nonzero__ =__bool__
+
     def is_equal_to(self, _right, absprec=None):
         r"""
         Returns whether self is equal to right modulo
diff --git a/src/sage/rings/padics/FM_template.pxi b/src/sage/rings/padics/FM_template.pxi
index 6a44d5cd491..22d644513e6 100644
--- a/src/sage/rings/padics/FM_template.pxi
+++ b/src/sage/rings/padics/FM_template.pxi
@@ -481,7 +481,7 @@ cdef class FMElement(pAdicTemplateElement):
         cdef long val = self.valuation_c()
         return mpz_cmp_si((<Integer>absprec).value, val) <= 0
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Returns True if this element is distinguishable from zero.
 
@@ -497,6 +497,8 @@ cdef class FMElement(pAdicTemplateElement):
         """
         return not ciszero(self.value, self.prime_pow)
 
+    __nonzero__ =__bool__
+
     def is_equal_to(self, _right, absprec=None):
         r"""
         Returns whether this element is equal to ``right`` modulo `p^{\mbox{absprec}}`.
diff --git a/src/sage/rings/polynomial/laurent_polynomial.pyx b/src/sage/rings/polynomial/laurent_polynomial.pyx
index 8e4ffc2e2c7..b6ffd69fbf7 100644
--- a/src/sage/rings/polynomial/laurent_polynomial.pyx
+++ b/src/sage/rings/polynomial/laurent_polynomial.pyx
@@ -161,7 +161,7 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
         """
         return self.__u.is_zero()
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Check if ``self`` is non-zero.
 
@@ -177,6 +177,8 @@ cdef class LaurentPolynomial_univariate(LaurentPolynomial_generic):
         """
         return not self.__u.is_zero()
 
+    __nonzero__ =__bool__
+
     def _im_gens_(self, codomain, im_gens):
         """
         Return the image of ``self`` under the morphism defined by
diff --git a/src/sage/rings/polynomial/multi_polynomial_element.py b/src/sage/rings/polynomial/multi_polynomial_element.py
index 5ca295c3652..8f769df553a 100644
--- a/src/sage/rings/polynomial/multi_polynomial_element.py
+++ b/src/sage/rings/polynomial/multi_polynomial_element.py
@@ -1464,7 +1464,7 @@ def __ne__(self,right):
             return CommutativeRingElement.__ne__(self, right)
         return self._MPolynomial_element__element != right._MPolynomial_element__element
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Returns True if self != 0
 
@@ -1474,6 +1474,8 @@ def __nonzero__(self):
         """
         return self._MPolynomial_element__element.dict()!={}
 
+    __nonzero__ =__bool__
+
     def _floordiv_(self, right):
         r"""
         Quotient of division of self by other. This is denoted //.
diff --git a/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx b/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
index 81f64bb3e57..5ff7f128af8 100644
--- a/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
+++ b/src/sage/rings/polynomial/multi_polynomial_libsingular.pyx
@@ -3888,7 +3888,7 @@ cdef class MPolynomial_libsingular(sage.rings.polynomial.multi_polynomial.MPolyn
         else:
             return False
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         EXAMPLES::
 
@@ -3903,6 +3903,8 @@ cdef class MPolynomial_libsingular(sage.rings.polynomial.multi_polynomial.MPolyn
         else:
             return False
 
+    __nonzero__ =__bool__
+
     cpdef _floordiv_(self, right):
         """
         Perform division with remainder and return the quotient.
diff --git a/src/sage/rings/polynomial/pbori.pyx b/src/sage/rings/polynomial/pbori.pyx
index 49c1a23213d..685514abd85 100644
--- a/src/sage/rings/polynomial/pbori.pyx
+++ b/src/sage/rings/polynomial/pbori.pyx
@@ -3532,7 +3532,7 @@ cdef class BooleanPolynomial(MPolynomial):
         """
         return self._pbpoly.isZero()
 
-    def __nonzero__(self):
+    def __bool__(self):
         r"""
         Check if ``self`` is zero.
 
@@ -3554,6 +3554,8 @@ cdef class BooleanPolynomial(MPolynomial):
         """
         return not self._pbpoly.isZero()
 
+    __nonzero__ =__bool__
+
     def is_one(BooleanPolynomial self):
         """
         Check if self is 1.
diff --git a/src/sage/rings/polynomial/plural.pyx b/src/sage/rings/polynomial/plural.pyx
index 83d632afc45..f048791bf59 100644
--- a/src/sage/rings/polynomial/plural.pyx
+++ b/src/sage/rings/polynomial/plural.pyx
@@ -2629,7 +2629,7 @@ cdef class NCPolynomial_plural(RingElement):
         else:
             return False
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         EXAMPLES::
 
@@ -2648,6 +2648,8 @@ cdef class NCPolynomial_plural(RingElement):
         else:
             return False
 
+    __nonzero__ =__bool__
+
 
 #####################################################################
 
diff --git a/src/sage/rings/polynomial/polynomial_element.pyx b/src/sage/rings/polynomial/polynomial_element.pyx
index 0a44c18ce87..0e4b1c89619 100644
--- a/src/sage/rings/polynomial/polynomial_element.pyx
+++ b/src/sage/rings/polynomial/polynomial_element.pyx
@@ -930,7 +930,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
             if c: return c
         return 0
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         EXAMPLES::
 
@@ -943,6 +943,8 @@ cdef class Polynomial(CommutativeAlgebraElement):
         """
         return self.degree() >= 0
 
+    __nonzero__ =__bool__
+
     def __getitem__(self, n):
         r"""
         Return the `n`-th coefficient of ``self``.
@@ -8976,7 +8978,7 @@ cdef class Polynomial_generic_dense(Polynomial):
         """
         return make_generic_polynomial, (self._parent, self.__coeffs)
 
-    def __nonzero__(self):
+    def __bool__(self):
         return len(self.__coeffs) > 0
 
     cdef int __normalize(self) except -1:
@@ -8986,7 +8988,7 @@ cdef class Polynomial_generic_dense(Polynomial):
         Check that exceptions are propagated correctly (:trac:`18274`)::
 
             sage: class BrokenRational(Rational):
-            ....:     def __nonzero__(self):
+            ....:     def __bool__(self):
             ....:         raise NotImplementedError("cannot check whether number is non-zero")
             sage: z = BrokenRational()
             sage: R.<x> = QQ[]
diff --git a/src/sage/rings/polynomial/polynomial_integer_dense_flint.pyx b/src/sage/rings/polynomial/polynomial_integer_dense_flint.pyx
index b143507c8e2..1f62a0f0b78 100644
--- a/src/sage/rings/polynomial/polynomial_integer_dense_flint.pyx
+++ b/src/sage/rings/polynomial/polynomial_integer_dense_flint.pyx
@@ -749,22 +749,24 @@ cdef class Polynomial_integer_dense_flint(Polynomial):
         """
         return (fmpz_poly_degree(self.__poly) == -1)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Check if self is not zero.
 
         EXAMPLES::
 
             sage: R.<x> = ZZ[]
-            sage: R(0).__nonzero__()
+            sage: bool(R(0))
             False
-            sage: R(1).__nonzero__()
+            sage: bool(R(1))
             True
-            sage: x.__nonzero__()
+            sage: bool(x)
             True
         """
         return not (fmpz_poly_degree(self.__poly) == -1)
 
+    __nonzero__ =__bool__
+
     @coerce_binop
     def gcd(self, right):
         r"""
diff --git a/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx b/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx
index 6517f1b6100..460026f8d9b 100644
--- a/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx
+++ b/src/sage/rings/polynomial/polynomial_modn_dense_ntl.pyx
@@ -1043,7 +1043,7 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
     def is_gen(self):
         return zz_pX_IsX(self.x)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         TESTS::
 
@@ -1056,6 +1056,8 @@ cdef class Polynomial_dense_modn_ntl_zz(Polynomial_dense_mod_n):
         """
         return not zz_pX_IsZero(self.x)
 
+    __nonzero__ =__bool__
+
     def valuation(self):
         """
         Returns the valuation of self, that is, the power of the
@@ -1604,7 +1606,7 @@ cdef class Polynomial_dense_modn_ntl_ZZ(Polynomial_dense_mod_n):
                 return n
         return infinity
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         TESTS::
 
diff --git a/src/sage/rings/polynomial/polynomial_rational_flint.pyx b/src/sage/rings/polynomial/polynomial_rational_flint.pyx
index b6c1b25e7cb..4a63dd57e98 100644
--- a/src/sage/rings/polynomial/polynomial_rational_flint.pyx
+++ b/src/sage/rings/polynomial/polynomial_rational_flint.pyx
@@ -712,7 +712,7 @@ cdef class Polynomial_rational_flint(Polynomial):
         """
         return fmpq_poly_is_one(self.__poly)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Returns whether or not self is non-zero.
 
@@ -727,6 +727,8 @@ cdef class Polynomial_rational_flint(Polynomial):
         """
         return not fmpq_poly_is_zero(self.__poly)
 
+    __nonzero__ =__bool__
+
     ###########################################################################
     # Shifting                                                                #
     ###########################################################################
diff --git a/src/sage/rings/polynomial/polynomial_template.pxi b/src/sage/rings/polynomial/polynomial_template.pxi
index 20b584f18e8..8cd2ba5838a 100644
--- a/src/sage/rings/polynomial/polynomial_template.pxi
+++ b/src/sage/rings/polynomial/polynomial_template.pxi
@@ -517,7 +517,7 @@ cdef class Polynomial_template(Polynomial):
             raise ValueError("Cannot coerce polynomial with degree %d to integer."%(self.degree()))
         return int(self[0])
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         EXAMPLE::
 
@@ -529,6 +529,8 @@ cdef class Polynomial_template(Polynomial):
         """
         return not celement_is_zero(&self.x, (<Polynomial_template>self)._cparent)
 
+    __nonzero__ =__bool__
+
     cpdef int _cmp_(left, right) except -2:
         """
         EXAMPLE::
diff --git a/src/sage/rings/polynomial/skew_polynomial_element.pyx b/src/sage/rings/polynomial/skew_polynomial_element.pyx
index 66fc5790115..a139662a7a0 100644
--- a/src/sage/rings/polynomial/skew_polynomial_element.pyx
+++ b/src/sage/rings/polynomial/skew_polynomial_element.pyx
@@ -1675,7 +1675,7 @@ cdef class SkewPolynomial(AlgebraElement):
         return (self.degree() == self.valuation() and
                 self.leading_coefficient()._is_atomic())
 
-    def __nonzero__(self):
+    def __bool__(self):
         r"""
         Test whether ``self`` is nonzero.
 
@@ -1685,14 +1685,16 @@ cdef class SkewPolynomial(AlgebraElement):
             sage: sigma = R.hom([t+1])
             sage: S.<x> = R['x',sigma]
             sage: a = x + 1
-            sage: a.__nonzero__()
+            sage: bool(a)
             True
             sage: b = S.zero()
-            sage: b.__nonzero__()
+            sage: bool(b)
             False
         """
         return not self.is_zero()
 
+    __nonzero__ =__bool__
+
     def base_ring(self):
         r"""
         Return the base ring of ``self``.
diff --git a/src/sage/rings/power_series_poly.pyx b/src/sage/rings/power_series_poly.pyx
index 3785a69f38a..5bec0b1434a 100644
--- a/src/sage/rings/power_series_poly.pyx
+++ b/src/sage/rings/power_series_poly.pyx
@@ -140,22 +140,24 @@ cdef class PowerSeries_poly(PowerSeries):
         """
         return self.__f.degree()
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return True if self is nonzero, and False otherwise.
 
         EXAMPLES::
 
             sage: R.<t> = GF(11)[[]]
-            sage: (1 + t + O(t^18)).__nonzero__()
+            sage: bool(1 + t + O(t^18))
             True
-            sage: R(0).__nonzero__()
+            sage: bool(R(0))
             False
-            sage: O(t^18).__nonzero__()
+            sage: bool(O(t^18))
             False
         """
         return not not self.__f
 
+    __nonzero__ =__bool__
+
     def __call__(self, *x, **kwds):
         """
         Evaluate the series at x=a.
diff --git a/src/sage/rings/power_series_ring_element.pyx b/src/sage/rings/power_series_ring_element.pyx
index 77c12ec97aa..8feec98b14e 100644
--- a/src/sage/rings/power_series_ring_element.pyx
+++ b/src/sage/rings/power_series_ring_element.pyx
@@ -906,7 +906,7 @@ cdef class PowerSeries(AlgebraElement):
         # endif
         return prec
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return True if this power series is not equal to 0.
 
@@ -926,6 +926,8 @@ cdef class PowerSeries(AlgebraElement):
         """
         return not not self.polynomial()
 
+    __nonzero__ =__bool__
+
     def is_unit(self):
         """
         Return True if this power series is invertible.
diff --git a/src/sage/rings/qqbar.py b/src/sage/rings/qqbar.py
index c0edcbe28df..636a0212a8b 100644
--- a/src/sage/rings/qqbar.py
+++ b/src/sage/rings/qqbar.py
@@ -3805,7 +3805,7 @@ def __ne__(self, other):
         """
         return not self == other
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Check whether self is equal is nonzero. This is fast if
         interval arithmetic proves that self is nonzero, but may be
@@ -3813,9 +3813,9 @@ def __nonzero__(self):
 
         EXAMPLES::
 
-            sage: (QQbar.zeta(2) + 1).__nonzero__()
+            sage: bool(QQbar.zeta(2) + 1)
             False
-            sage: (QQbar.zeta(7) / (2^500)).__nonzero__()
+            sage: bool(QQbar.zeta(7) / (2^500))
             True
         """
         val = self._value
@@ -3832,7 +3832,9 @@ def __nonzero__(self):
 
         # Sigh...
         self.exactify()
-        return self.__nonzero__()
+        return self.__bool__()
+
+    __nonzero__ =__bool__
 
     def __pow__(self, e):
         r""" ``self**p`` returns the `p`'th power of self (where `p` can
diff --git a/src/sage/rings/quotient_ring_element.py b/src/sage/rings/quotient_ring_element.py
index 8e59d6d114d..a2eadebf625 100644
--- a/src/sage/rings/quotient_ring_element.py
+++ b/src/sage/rings/quotient_ring_element.py
@@ -126,7 +126,7 @@ def lift(self):
         """
         return self.__rep
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return True if quotient ring element is non-zero in the
         quotient ring `R/I`, by determining whether the element
@@ -143,13 +143,13 @@ def __nonzero__(self):
 
         TESTS::
 
-            sage: S(0).__nonzero__()
-            False
-            sage: (a-a).__nonzero__()
+            sage: bool(a - a)
             False
         """
         return self.__rep not in self.parent().defining_ideal()
 
+    __nonzero__ =__bool__
+
     def is_unit(self):
         """
         Return True if self is a unit in the quotient ring.
diff --git a/src/sage/rings/rational.pyx b/src/sage/rings/rational.pyx
index 25c23ac9d99..08da03bb94c 100644
--- a/src/sage/rings/rational.pyx
+++ b/src/sage/rings/rational.pyx
@@ -16,7 +16,7 @@ AUTHORS:
 - David Harvey (2006-09-15): added nth_root
 
 - Pablo De Napoli (2007-04-01): corrected the implementations of
-  multiplicative_order, is_one; optimized __nonzero__ ; documented:
+  multiplicative_order, is_one; optimized __bool__ ; documented:
   lcm,gcd
 
 - John Cremona (2009-05-15): added support for local and global
@@ -2551,7 +2551,7 @@ cdef class Rational(sage.structure.element.FieldElement):
         mpq_neg(x.value, self.value)
         return x
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` if this rational number is nonzero.
 
@@ -2559,9 +2559,7 @@ cdef class Rational(sage.structure.element.FieldElement):
 
         EXAMPLES::
 
-            sage: (-4/17).__nonzero__()
-            True
-            sage: (0/5).__nonzero__()
+            sage: bool(0/5)
             False
             sage: bool(-4/17)
             True
@@ -2569,6 +2567,8 @@ cdef class Rational(sage.structure.element.FieldElement):
         # A rational number is zero iff its numerator is zero.
         return mpq_sgn(self.value) != 0
 
+    __nonzero__ =__bool__
+
     def __abs__(self):
         """
         Return the absolute value of this rational number.
diff --git a/src/sage/rings/real_arb.pyx b/src/sage/rings/real_arb.pyx
index 7b42a18e89a..391c0dd477c 100644
--- a/src/sage/rings/real_arb.pyx
+++ b/src/sage/rings/real_arb.pyx
@@ -2006,7 +2006,7 @@ cdef class RealBall(RingElement):
         """
         return arb_is_nonzero(self.value)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` iff this ball is not the zero ball, i.e. if it its
         midpoint and radius are not both zero.
@@ -2031,6 +2031,8 @@ cdef class RealBall(RingElement):
         """
         return not arb_is_zero(self.value)
 
+    __nonzero__ =__bool__
+
     def is_exact(self):
         """
         Return ``True`` iff the radius of this ball is zero.
diff --git a/src/sage/rings/real_interval_absolute.pyx b/src/sage/rings/real_interval_absolute.pyx
index ec9f72afc39..0a1479c13df 100644
--- a/src/sage/rings/real_interval_absolute.pyx
+++ b/src/sage/rings/real_interval_absolute.pyx
@@ -567,7 +567,7 @@ cdef class RealIntervalAbsoluteElement(FieldElement):
     cdef bint is_exact(self):
         return not self._diameter
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` for anything except exact zero.
 
@@ -584,6 +584,8 @@ cdef class RealIntervalAbsoluteElement(FieldElement):
         """
         return not not self._mantissa or not not self._diameter
 
+    __nonzero__ =__bool__
+
     def __neg__(self):
         """
         TESTS::
diff --git a/src/sage/rings/real_lazy.pyx b/src/sage/rings/real_lazy.pyx
index 26542878f33..caa30177ccb 100644
--- a/src/sage/rings/real_lazy.pyx
+++ b/src/sage/rings/real_lazy.pyx
@@ -988,7 +988,7 @@ cdef class LazyWrapper(LazyFieldElement):
         """
         return <double>self._value
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Check to see if ``self`` is not zero.
 
@@ -1002,6 +1002,8 @@ cdef class LazyWrapper(LazyFieldElement):
         """
         return not not self._value
 
+    __nonzero__ =__bool__
+
     def __hash__(self):
         """
         Return the hash value of ``self``.
diff --git a/src/sage/rings/real_mpfi.pyx b/src/sage/rings/real_mpfi.pyx
index 3357b958fb3..259bded4c93 100644
--- a/src/sage/rings/real_mpfi.pyx
+++ b/src/sage/rings/real_mpfi.pyx
@@ -3731,25 +3731,27 @@ cdef class RealIntervalFieldElement(RingElement):
         elif op == 5: #>=
             return mpfr_lessequal_p(&rt.value.right, &lt.value.left)
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` if ``self`` is not known to be exactly zero.
 
         EXAMPLES::
 
-            sage: RIF(0).__nonzero__()
+            sage: bool(RIF(0))
             False
-            sage: RIF(1).__nonzero__()
+            sage: bool(RIF(1))
             True
-            sage: RIF(1, 2).__nonzero__()
+            sage: bool(RIF(1, 2))
             True
-            sage: RIF(0, 1).__nonzero__()
+            sage: bool(RIF(0, 1))
             True
-            sage: RIF(-1, 1).__nonzero__()
+            sage: bool(RIF(-1, 1))
             True
         """
         return not (mpfr_zero_p(&self.value.left) and mpfr_zero_p(&self.value.right))
 
+    __nonzero__ =__bool__
+
     cpdef int _cmp_(left, right) except -2:
         """
         Compare two intervals lexicographically.
diff --git a/src/sage/rings/real_mpfr.pyx b/src/sage/rings/real_mpfr.pyx
index ce26f34475a..a2388fc63b3 100644
--- a/src/sage/rings/real_mpfr.pyx
+++ b/src/sage/rings/real_mpfr.pyx
@@ -3784,30 +3784,32 @@ cdef class RealNumber(sage.structure.element.RingElement):
         """
         return mpfr_integer_p(self.value) != 0
 
-    def __nonzero__(self):
+    def __bool__(self):
         """
         Return ``True`` if ``self`` is nonzero.
 
         EXAMPLES::
 
-            sage: RR(1).__nonzero__()
+            sage: bool(RR(1))
             True
-            sage: RR(0).__nonzero__()
+            sage: bool(RR(0))
             False
-            sage: RR('inf').__nonzero__()
+            sage: bool(RR('inf'))
             True
 
         TESTS:
 
         Check that :trac:`20502` is fixed::
 
-            sage: RR('nan').__nonzero__()
+            sage: bool(RR('nan'))
             True
             sage: RR('nan').is_zero()
             False
         """
         return not mpfr_zero_p(self.value)
 
+    __nonzero__ =__bool__
+
     cpdef int _cmp_(left, right) except -2:
         """
         Return ``-1`` if exactly one of the numbers is ``NaN``.  Return ``-1``
diff --git a/src/sage/rings/universal_cyclotomic_field.py b/src/sage/rings/universal_cyclotomic_field.py
index a94af63f105..19f5d5679a4 100644
--- a/src/sage/rings/universal_cyclotomic_field.py
+++ b/src/sage/rings/universal_cyclotomic_field.py
@@ -275,7 +275,7 @@ def __init__(self, parent, obj):
         self._obj = obj
         FieldElement.__init__(self, parent)
 
-    def __nonzero__(self):
+    def __bool__(self):
         r"""
         TESTS::
 
@@ -285,6 +285,8 @@ def __nonzero__(self):
         """
         return bool(self._obj)
 
+    __nonzero__ =__bool__
+
     def __reduce__(self):
         r"""
         TESTS::