diff --git a/src/sage/rings/complex_arb.pyx b/src/sage/rings/complex_arb.pyx
index 0cab5ba87e9..cdd0a688aa5 100644
--- a/src/sage/rings/complex_arb.pyx
+++ b/src/sage/rings/complex_arb.pyx
@@ -360,7 +360,7 @@ class ComplexBallField(UniqueRepresentation, sage.rings.abc.ComplexBallField):
         """
         return super().__classcall__(cls, precision)
 
-    def __init__(self, long precision=53):
+    def __init__(self, long precision=53) -> None:
         r"""
         Initialize the complex ball field.
 
@@ -1399,7 +1399,7 @@ cdef class ComplexBall(RingElement):
         """
         acb_clear(self.value)
 
-    def __init__(self, parent, x=None, y=None):
+    def __init__(self, parent, x=None, y=None) -> None:
         """
         Initialize the :class:`ComplexBall`.
 
@@ -1529,7 +1529,7 @@ cdef class ComplexBall(RingElement):
         else:
             raise TypeError("unsupported initializer")
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         TESTS::
 
@@ -2333,7 +2333,7 @@ cdef class ComplexBall(RingElement):
         return (arb_is_nonzero(acb_realref(self.value))
                 or arb_is_nonzero(acb_imagref(self.value)))
 
-    def __bool__(self):
+    def __bool__(self) -> bool:
         """
         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.
@@ -2590,7 +2590,7 @@ cdef class ComplexBall(RingElement):
             if _do_sig(prec(self)): sig_off()
         return res
 
-    def __contains__(self, other):
+    def __contains__(self, other) -> bool:
         """
         Return ``True`` if ``other`` can be verified to be contained in ``self``.
 
diff --git a/src/sage/rings/complex_interval.pyx b/src/sage/rings/complex_interval.pyx
index b0b9f14263b..6ef90638d3b 100644
--- a/src/sage/rings/complex_interval.pyx
+++ b/src/sage/rings/complex_interval.pyx
@@ -125,7 +125,7 @@ cdef class ComplexIntervalFieldElement(FieldElement):
         mpfi_init2(self.__re, self._prec)
         mpfi_init2(self.__im, self._prec)
 
-    def __init__(self, parent, real, imag=None, int base=10):
+    def __init__(self, parent, real, imag=None, int base=10) -> None:
         """
         Initialize a complex number (interval).
 
@@ -168,7 +168,7 @@ cdef class ComplexIntervalFieldElement(FieldElement):
         """
         return self.str(10)
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash value of ``self``.
 
@@ -664,7 +664,7 @@ cdef class ComplexIntervalFieldElement(FieldElement):
 
         return center
 
-    def __contains__(self, other):
+    def __contains__(self, other) -> bool:
         """
         Test whether ``other`` is totally contained in ``self``.
 
@@ -1406,7 +1406,7 @@ cdef class ComplexIntervalFieldElement(FieldElement):
         return complex(self.real().n(self._prec),
                        self.imag().n(self._prec))
 
-    def __bool__(self):
+    def __bool__(self) -> bool:
         """
         Return ``True`` if ``self`` is not known to be exactly zero.
 
diff --git a/src/sage/rings/ideal.py b/src/sage/rings/ideal.py
index 589962e599d..eacdda6dd2f 100644
--- a/src/sage/rings/ideal.py
+++ b/src/sage/rings/ideal.py
@@ -31,7 +31,7 @@
 from sage.categories.rings import Rings
 from sage.categories.fields import Fields
 from sage.structure.element import MonoidElement
-from sage.structure.richcmp import rich_to_bool, richcmp, op_NE
+from sage.structure.richcmp import rich_to_bool, op_NE
 from sage.structure.sequence import Sequence
 
 
@@ -253,7 +253,7 @@ class Ideal_generic(MonoidElement):
 
     See :func:`Ideal()`.
     """
-    def __init__(self, ring, gens, coerce=True, **kwds):
+    def __init__(self, ring, gens, coerce=True, **kwds) -> None:
         """
         Initialize this ideal.
 
@@ -318,7 +318,7 @@ def _repr_short(self):
             s = repr(x)
             if '\n' in s:
                 has_return = True
-                s = s.replace('\n','\n  ')
+                s = s.replace('\n', '\n  ')
             L.append(s)
         if has_return:
             return '\n(\n  %s\n)\n' % (',\n\n  '.join(L))
@@ -395,7 +395,7 @@ def _richcmp_(self, other, op):
             return rich_to_bool(op, +1)
         raise NotImplementedError(f'ideal comparison in {self.ring()} is not implemented')
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Check if ``x`` is in ``self``.
 
@@ -439,7 +439,7 @@ def _contains_(self, x):
         """
         raise NotImplementedError
 
-    def __bool__(self):
+    def __bool__(self) -> bool:
         r"""
         Return ``True`` if this ideal is not `(0)`.
 
@@ -1081,7 +1081,7 @@ def __mul__(self, other):
             try:
                 if self.ring().has_coerce_map_from(other):
                     return self
-            except (TypeError,ArithmeticError,ValueError):
+            except (TypeError, ArithmeticError, ValueError):
                 pass
             other = self.ring().ideal(other)
         return self._mul_(other)
@@ -1107,7 +1107,8 @@ def _mul_(self, other):
             sage: I._mul_(J)
             Ideal (x^3*y, x^2*y^2, x^3*z, x^2*y*z, x^4, x^3*y) of Multivariate Polynomial Ring in x, y, z over Rational Field
         """
-        return self.ring().ideal([z for z in [x*y for x in self.gens() for y in other.gens()] if z])
+        return self.ring().ideal([z for x in self.gens() for y in other.gens()
+                                  if (z := x * y)])
 
     def __rmul__(self, other):
         """
@@ -1124,10 +1125,11 @@ def __rmul__(self, other):
             try:
                 if self.ring().has_coerce_map_from(other):
                     return self
-            except (TypeError,ArithmeticError,ValueError):
+            except (TypeError, ArithmeticError, ValueError):
                 pass
             other = self.ring().ideal(other)
-        return self.ring().ideal([z for z in [y*x for x in self.gens() for y in other.gens()] if z])
+        return self.ring().ideal([z for x in self.gens() for y in other.gens()
+                                  if (z := y * x)])
 
     def norm(self):
         """
@@ -1284,7 +1286,7 @@ class Ideal_principal(Ideal_generic):
     See :func:`Ideal()`.
     """
     # now Ideal_principal takes a list.
-    #def __init__(self, ring, gen):
+    # def __init__(self, ring, gen):
     #    Ideal_generic.__init__(self, ring, [gen])
 
     def _repr_(self):
@@ -1366,7 +1368,7 @@ def gen(self, i=0):
             raise ValueError(f"i (={i}) must be 0")
         return self.gens()[0]
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         """
         Return ``True`` if ``x`` is in ``self``.
 
@@ -1388,7 +1390,7 @@ def __contains__(self, x):
         except NotImplementedError:
             return self._contains_(self.ring()(x))
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         r"""
         Very stupid constant hash function!
 
@@ -1622,10 +1624,10 @@ def is_prime(self):
             sage: RR.ideal(7).is_prime()
             False
         """
-        if self.is_zero(): # PIDs are integral domains by definition
+        if self.is_zero():  # PIDs are integral domains by definition
             return True
         g = self.gen()
-        if g.is_one():     # The ideal (1) is never prime
+        if g.is_one():      # The ideal (1) is never prime
             return False
         if hasattr(g, 'is_irreducible'):
             return g.is_irreducible()
@@ -1819,7 +1821,7 @@ def Cyclic(R, n=None, homog=False, singular=None):
     if not homog:
         I = singular.cyclic(n)
     else:
-        I = singular.cyclic(n).homog(R2.gen(n-1))
+        I = singular.cyclic(n).homog(R2.gen(n - 1))
     return R2.ideal(I).change_ring(R)
 
 
@@ -1870,7 +1872,7 @@ def Katsura(R, n=None, homog=False, singular=None):
     if not homog:
         I = singular.katsura(n)
     else:
-        I = singular.katsura(n).homog(R2.gen(n-1))
+        I = singular.katsura(n).homog(R2.gen(n - 1))
     return R2.ideal(I).change_ring(R)
 
 
diff --git a/src/sage/rings/polynomial/laurent_polynomial_ideal.py b/src/sage/rings/polynomial/laurent_polynomial_ideal.py
index 096e6bff57d..a32d9ef506e 100644
--- a/src/sage/rings/polynomial/laurent_polynomial_ideal.py
+++ b/src/sage/rings/polynomial/laurent_polynomial_ideal.py
@@ -28,7 +28,7 @@
 
 
 class LaurentPolynomialIdeal( Ideal_generic ):
-    def __init__(self, ring, gens, coerce=True, hint=None):
+    def __init__(self, ring, gens, coerce=True, hint=None) -> None:
         r"""
         Create an ideal in a Laurent polynomial ring.
 
@@ -184,7 +184,7 @@ def _richcmp_(self, right_r, op):
         else:
             raise ValueError("invalid comparison")
 
-    def __contains__(self, f):
+    def __contains__(self, f) -> bool:
         """
         Implement containment testing (in) for Laurent polynomial ideals.
 
diff --git a/src/sage/rings/polynomial/pbori/pbori.pyx b/src/sage/rings/polynomial/pbori/pbori.pyx
index d5bc16bbe11..88e4095678a 100644
--- a/src/sage/rings/polynomial/pbori/pbori.pyx
+++ b/src/sage/rings/polynomial/pbori/pbori.pyx
@@ -325,7 +325,7 @@ cdef class BooleanPolynomialRing(BooleanPolynomialRing_base):
         sage: P == S
         False
     """
-    def __init__(self, n=None, names=None, order='lex'):
+    def __init__(self, n=None, names=None, order='lex') -> None:
         """
         Create a new boolean polynomial ring.
 
@@ -1016,7 +1016,7 @@ cdef class BooleanPolynomialRing(BooleanPolynomialRing_base):
         else:
             return self._zero_element
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return a hash of this boolean polynomial ring.
 
@@ -1877,7 +1877,7 @@ class BooleanMonomialMonoid(UniqueRepresentation, Monoid_class):
         True
         sage: TestSuite(M).run()
     """
-    def __init__(self, BooleanPolynomialRing polring):
+    def __init__(self, BooleanPolynomialRing polring) -> None:
         """
         Create a new boolean polynomial ring.
 
@@ -1915,7 +1915,7 @@ class BooleanMonomialMonoid(UniqueRepresentation, Monoid_class):
         """
         return "MonomialMonoid of %s" % (str(self._ring))
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return a hash for this monoid.
 
@@ -2229,7 +2229,7 @@ cdef class BooleanMonomial(MonoidElement):
        Use the :meth:`BooleanMonomialMonoid__call__` method and not
        this constructor to construct these objects.
     """
-    def __init__(self, parent):
+    def __init__(self, parent) -> None:
         """
         EXAMPLES::
 
@@ -2363,7 +2363,7 @@ cdef class BooleanMonomial(MonoidElement):
             res *= d[var]
         return res
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return a hash of this monomial.
 
@@ -2571,7 +2571,7 @@ cdef class BooleanMonomial(MonoidElement):
         """
         return new_BS_from_PBSet(self._pbmonom.set(), self._ring)
 
-    def __len__(BooleanMonomial self):
+    def __len__(BooleanMonomial self) -> int:
         """
         Return 1.
 
@@ -2940,7 +2940,7 @@ cdef class BooleanPolynomial(MPolynomial):
         Do not use this method to construct boolean polynomials, but
         use the appropriate ``__call__`` method in the parent.
     """
-    def __init__(self, parent):
+    def __init__(self, parent) -> None:
         self._parent = parent
         self._pbpoly = PBBoolePolynomial((<BooleanPolynomialRing?>parent)._pbring)
 
@@ -3487,7 +3487,7 @@ cdef class BooleanPolynomial(MPolynomial):
         """
         return self._pbpoly.isZero()
 
-    def __bool__(self):
+    def __bool__(self) -> bool:
         r"""
         Check if ``self`` is not zero.
 
@@ -3845,7 +3845,7 @@ cdef class BooleanPolynomial(MPolynomial):
         else:
             return B._base._zero_element
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         r"""
         Return hash for ``self``.
 
@@ -4799,7 +4799,7 @@ cdef inline BooleanPolynomialIterator new_BPI_from_BooleanPolynomial(BooleanPoly
 
 
 class BooleanPolynomialIdeal(MPolynomialIdeal):
-    def __init__(self, ring, gens=[], coerce=True):
+    def __init__(self, ring, gens=[], coerce=True) -> None:
         """
         Construct an ideal in the boolean polynomial ring.
 
@@ -5145,7 +5145,7 @@ class BooleanPolynomialIdeal(MPolynomialIdeal):
         """
         return self.basis.reduced()
 
-    def __eq__(self, other):
+    def __eq__(self, other) -> bool:
         """
         EXAMPLES::
 
@@ -5171,7 +5171,7 @@ class BooleanPolynomialIdeal(MPolynomialIdeal):
         else:
             return self.groebner_basis() == other.groebner_basis()
 
-    def __ne__(self, other):
+    def __ne__(self, other) -> bool:
         """
         EXAMPLES::
 
@@ -5274,7 +5274,7 @@ cdef class BooleSet:
 
       :class:`BooleSet` prints as ``{}`` but are not Python dictionaries.
     """
-    def __init__(self, param=None, ring=None):
+    def __init__(self, param=None, ring=None) -> None:
         cdef BooleanPolynomial p
         if isinstance(param, CCuddNavigator):
             if ring is None:
@@ -5309,7 +5309,7 @@ cdef class BooleSet:
             self._pbset = PBBooleSet((<BooleanPolynomial>p)._pbpoly)
             self._ring = detected_ring
 
-    def __repr__(self):
+    def __repr__(self) -> str:
         """
         EXAMPLES::
 
@@ -5579,7 +5579,7 @@ cdef class BooleSet:
         """
         return self._pbset.size()
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         EXAMPLES::
 
@@ -5615,7 +5615,7 @@ cdef class BooleSet:
         """
         return mod_mon_set(self, vs)
 
-    def __contains__(self, BooleanMonomial m):
+    def __contains__(self, BooleanMonomial m) -> bool:
         """
         Return ``True`` if ``m`` is in this set.
 
@@ -5931,7 +5931,7 @@ cdef class CCuddNavigator:
     def terminal_one(self):
         return self._pbnav.isTerminated()
 
-    def __richcmp__(CCuddNavigator self, CCuddNavigator other, int op):
+    def __richcmp__(CCuddNavigator self, CCuddNavigator other, int op) -> bool:
         """
         ::
 
@@ -5953,7 +5953,7 @@ cdef class CCuddNavigator:
         else:
             return NotImplemented
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         return self._pbnav.hash()
 
 
@@ -5972,7 +5972,7 @@ cdef class BooleanPolynomialVector:
         sage: all(vi.parent() is B for vi in v)
         True
     """
-    def __init__(self, I=None):
+    def __init__(self, I=None) -> None:
         """
         Create a new :class:`BooleanPolynomialVector`.
 
@@ -6154,7 +6154,7 @@ cdef class ReductionStrategy:
     """
     Functions and options for boolean polynomial reduction.
     """
-    def __init__(self, ring):
+    def __init__(self, ring) -> None:
         """
         EXAMPLES::
 
@@ -6423,7 +6423,7 @@ cdef class ReductionStrategy:
 
 
 cdef class BooleanPolynomialEntry:
-    def __init__(self, p):
+    def __init__(self, p) -> None:
         self.p = <BooleanPolynomial?>p
 
 
@@ -6433,7 +6433,7 @@ cdef class FGLMStrategy:
     Groebner basis with respect to a term ordering A to another
     Groebner basis with respect to a term ordering B.
     """
-    def __init__(self, from_ring, to_ring, BooleanPolynomialVector vec):
+    def __init__(self, from_ring, to_ring, BooleanPolynomialVector vec) -> None:
         """
         Execute the FGLM algorithm.
 
@@ -6508,7 +6508,7 @@ cdef class GroebnerStrategy:
 
       This class is mainly used internally.
     """
-    def __init__(self, param):
+    def __init__(self, param) -> None:
         """
         INPUT:
 
@@ -7013,7 +7013,7 @@ cdef inline CCuddNavigator new_CN_from_PBNavigator(PBNavigator juice,
 
 cdef class VariableBlock:
     def __init__(self, int size, int start_index, int offset, bint reverse,
-                 BooleanPolynomialRing ring):
+                 BooleanPolynomialRing ring) -> None:
         self._ring = ring
         self._block = new PBVarBlock(size, start_index, offset, reverse,
                                      ring._pbring)
@@ -7764,7 +7764,7 @@ def unpickle_BooleanPolynomialRing(n, names, order):
 
 
 cdef class BooleConstant:
-    def __init__(self, int value):
+    def __init__(self, int value) -> None:
         """
         Construct a boolean constant (modulo 2) from integer value:
 
@@ -7780,7 +7780,7 @@ cdef class BooleConstant:
         """
         self._pbconst = PBConstant(value)
 
-    def __repr__(self):
+    def __repr__(self) -> str:
         """
         EXAMPLES::
 
@@ -7908,7 +7908,7 @@ cdef class VariableFactory:
     """Implements PolyBoRi's ``Variable()`` constructor and
     a variable factory for given ring """
 
-    def __init__(self, BooleanPolynomialRing ring=None):
+    def __init__(self, BooleanPolynomialRing ring=None) -> None:
         """
         Initialize variable factory, if ring is given.
         Otherwise it initializes a plain constructor
@@ -7964,7 +7964,7 @@ cdef class MonomialFactory:
             sage: fac = MonomialFactory()
             sage: fac = MonomialFactory(B)
     """
-    def __init__(self, BooleanPolynomialRing ring=None):
+    def __init__(self, BooleanPolynomialRing ring=None) -> None:
         """
         Initialized a polynomial factory of ring is given.
         Otherwise it initializes a plain constructor.
@@ -8030,7 +8030,7 @@ cdef class PolynomialFactory:
     Implement PolyBoRi's ``Polynomial()`` constructor and
     a polynomial factory for given rings.
     """
-    def __init__(self, BooleanPolynomialRing ring=None):
+    def __init__(self, BooleanPolynomialRing ring=None) -> None:
         """
         Construct a polynomial factory if ring is given,
         or plain constructor otherwise.
diff --git a/src/sage/rings/polynomial/polydict.pyx b/src/sage/rings/polynomial/polydict.pyx
index 7d31fab7d56..6911cfa8eb8 100644
--- a/src/sage/rings/polynomial/polydict.pyx
+++ b/src/sage/rings/polynomial/polydict.pyx
@@ -107,7 +107,9 @@ cdef class PolyDict:
     can use the method :meth:`remove_zeros` which can be parametrized by a zero
     test.
     """
-    def __init__(self, pdict, zero=None, remove_zero=None, force_int_exponents=None, force_etuples=None, bint check=True):
+    def __init__(self, pdict, zero=None, remove_zero=None,
+                 force_int_exponents=None, force_etuples=None,
+                 bint check=True) -> None:
         """
         INPUT:
 
@@ -296,7 +298,7 @@ cdef class PolyDict:
         deprecation(34000, 'coerce_cefficients is deprecated; use apply_map instead')
         self.apply_map(A.coerce)
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         Return the hash.
 
@@ -313,7 +315,7 @@ cdef class PolyDict:
         """
         return hash(frozenset(self.__repn.items()))
 
-    def __bool__(self):
+    def __bool__(self) -> bool:
         """
         Return whether the PolyDict is empty.
 
@@ -326,7 +328,7 @@ cdef class PolyDict:
         """
         return bool(self.__repn)
 
-    def __len__(self):
+    def __len__(self) -> int:
         """
         Return the number of terms of this polynomial.
 
@@ -339,7 +341,7 @@ cdef class PolyDict:
         """
         return len(self.__repn)
 
-    def __richcmp__(PolyDict left, PolyDict right, int op):
+    def __richcmp__(PolyDict left, PolyDict right, int op) -> bool:
         """
         Implement the ``__richcmp__`` protocol for `PolyDict`s.
 
@@ -523,7 +525,7 @@ cdef class PolyDict:
         """
         return self.__repn.get(e, default)
 
-    def __repr__(self):
+    def __repr__(self) -> str:
         r"""
         String representation.
         """
@@ -1451,7 +1453,7 @@ cdef class ETuple:
         x._length = self._length
         return x
 
-    def __init__(self, data=None, length=None):
+    def __init__(self, data=None, length=None) -> None:
         """
         - ``ETuple()`` -> an empty ETuple
         - ``ETuple(sequence)`` -> ETuple initialized from sequence's items
@@ -1521,7 +1523,7 @@ cdef class ETuple:
         if self._data != <int*>0:
             sig_free(self._data)
 
-    def __bool__(self):
+    def __bool__(self) -> bool:
         r"""
         Return whether ``self`` is nonzero.
 
@@ -1663,7 +1665,7 @@ cdef class ETuple:
             return self._data[2 * ind + 1]
         return 0
 
-    def __hash__(self):
+    def __hash__(self) -> int:
         """
         x.__hash__() <==> hash(x)
         """
@@ -1688,7 +1690,7 @@ cdef class ETuple:
         """
         return self._length
 
-    def __contains__(self, elem):
+    def __contains__(self, elem) -> bool:
         """
         ``x.__contains__(n) <==> n in x``.
 
@@ -1712,7 +1714,7 @@ cdef class ETuple:
                 return True
         return False
 
-    def __richcmp__(ETuple self, ETuple other, op):
+    def __richcmp__(ETuple self, ETuple other, op) -> bool:
         """
         EXAMPLES::
 
@@ -1819,10 +1821,10 @@ cdef class ETuple:
             else:
                 yield 0
 
-    def __str__(self):
+    def __str__(self) -> str:
         return repr(self)
 
-    def __repr__(self):
+    def __repr__(self) -> str:
         r"""
         TESTS::
 
diff --git a/src/sage/rings/power_series_ring.py b/src/sage/rings/power_series_ring.py
index 129477ccc8c..6730f61806a 100644
--- a/src/sage/rings/power_series_ring.py
+++ b/src/sage/rings/power_series_ring.py
@@ -496,7 +496,7 @@ class PowerSeriesRing_generic(UniqueRepresentation, Parent, Nonexact):
     """
 
     def __init__(self, base_ring, name=None, default_prec=None, sparse=False,
-                 implementation=None, category=None):
+                 implementation=None, category=None) -> None:
         """
         Initialize a power series ring.
 
@@ -1239,7 +1239,7 @@ def random_element(self, prec=None, *args, **kwds):
             prec = self.default_prec()
         return self(self.__poly_ring.random_element(prec-1, *args, **kwds), prec)
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> None:
         """
         Return ``True`` if x is an element of this power series ring or
         canonically coerces to this ring.
diff --git a/src/sage/rings/species.py b/src/sage/rings/species.py
index 2e9ccf6bab2..4409a3d7de8 100644
--- a/src/sage/rings/species.py
+++ b/src/sage/rings/species.py
@@ -267,7 +267,7 @@ class of subgroups conjugate to ``C``, together with the
         lookup.append(elm)
         return elm
 
-    def __init__(self, parent, dis, domain_partition):
+    def __init__(self, parent, dis, domain_partition) -> None:
         r"""
         Initialize an atomic species.
 
@@ -324,7 +324,7 @@ def grade(self):
         S = self.parent().grading_set()
         return S(self._mc)
 
-    def __lt__(self, other):
+    def __lt__(self, other) -> bool:
         r"""
         Return whether ``self`` is less than ``other``.
 
@@ -379,7 +379,7 @@ def __lt__(self, other):
         H = libgap.ConjugateGroup(other._dis, conj_other)
         return GAP_FAIL != libgap.ContainedConjugates(S, G, H, True)
 
-    def __le__(self, other):
+    def __le__(self, other) -> bool:
         r"""
         Return whether ``self`` is less than or equal to ``other``.
 
@@ -537,7 +537,7 @@ def __classcall__(cls, names):
         names = normalize_names(-1, names)
         return super().__classcall__(cls, names)
 
-    def __init__(self, names):
+    def __init__(self, names) -> None:
         r"""
         Initialize the class of atomic species.
 
@@ -753,7 +753,7 @@ def _rename(self, n):
 
         _atomic_set_like_species(n, self._names)
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         r"""
         Return whether ``x`` is in ``self``.
 
@@ -1030,7 +1030,7 @@ def __classcall__(cls, names):
         names = normalize_names(-1, names)
         return UniqueRepresentation.__classcall__(cls, names)
 
-    def __init__(self, names):
+    def __init__(self, names) -> None:
         r"""
         Initialize the monoid of molecular species.
 
@@ -2404,7 +2404,7 @@ def __classcall__(cls, base_ring, names):
         names = normalize_names(-1, names)
         return super().__classcall__(cls, base_ring, names)
 
-    def __init__(self, base_ring, names):
+    def __init__(self, base_ring, names) -> None:
         r"""
         Initialize the ring of polynomial species.
 
diff --git a/src/sage/rings/valuation/valuation_space.py b/src/sage/rings/valuation/valuation_space.py
index 957a15ae488..399c8bdec22 100644
--- a/src/sage/rings/valuation/valuation_space.py
+++ b/src/sage/rings/valuation/valuation_space.py
@@ -100,7 +100,7 @@ class DiscretePseudoValuationSpace(UniqueRepresentation, Homset):
 
         sage: TestSuite(H).run() # long time
     """
-    def __init__(self, domain):
+    def __init__(self, domain) -> None:
         r"""
         TESTS::
 
@@ -187,7 +187,7 @@ def _repr_(self):
         """
         return "Discrete pseudo-valuations on %r" % (self.domain(),)
 
-    def __contains__(self, x):
+    def __contains__(self, x) -> bool:
         r"""
         Return whether ``x`` is a valuation in this space.