diff --git a/src/sage/algebras/lie_algebras/lie_algebra.py b/src/sage/algebras/lie_algebras/lie_algebra.py
index 11ebbeec026..1aa4e6e1782 100644
--- a/src/sage/algebras/lie_algebras/lie_algebra.py
+++ b/src/sage/algebras/lie_algebras/lie_algebra.py
@@ -835,12 +835,12 @@ def gens(self) -> tuple:
             (x, y)
         """
         G = self.lie_algebra_generators()
+        if not self.variable_names():
+            return tuple(G)
         try:
             return tuple(G[i] for i in self.variable_names())
         except (KeyError, IndexError):
             return tuple(G[i] for i in self.indices())
-        except ValueError:
-            return tuple(G)
 
     def gen(self, i):
         """
diff --git a/src/sage/algebras/lie_algebras/quotient.py b/src/sage/algebras/lie_algebras/quotient.py
index 4fec104335d..d0b83fbca88 100644
--- a/src/sage/algebras/lie_algebras/quotient.py
+++ b/src/sage/algebras/lie_algebras/quotient.py
@@ -227,13 +227,17 @@ def __classcall_private__(cls, ambient, I, names=None, index_set=None,
             index_set = [i[0] for i in index_set_mapping if i[0] not in I_supp]
 
         if names is None:
-            try:
-                amb_names = dict(zip([i[1] for i in index_set_mapping], ambient.variable_names()))
-                names = [amb_names[i] for i in index_set]
-            except (ValueError, KeyError):
-                # ambient has not assigned variable names
-                # or the names are for the generators rather than the basis
-                names = 'e'
+            names = 'e'
+            if ambient.variable_names():
+                # ambient has not assigned variable names...
+                try:
+                    amb_names = dict(zip([i[1] for i in index_set_mapping],
+                                         ambient.variable_names()))
+                    names = [amb_names[i] for i in index_set]
+                except KeyError:
+                    # or the names are for the generators rather than
+                    # the basis.
+                    pass
         if isinstance(names, str):
             if len(index_set) == 1:
                 names = [names]
diff --git a/src/sage/algebras/lie_conformal_algebras/lie_conformal_algebra_element.py b/src/sage/algebras/lie_conformal_algebras/lie_conformal_algebra_element.py
index 27796941d2b..7fb1b373c13 100644
--- a/src/sage/algebras/lie_conformal_algebras/lie_conformal_algebra_element.py
+++ b/src/sage/algebras/lie_conformal_algebras/lie_conformal_algebra_element.py
@@ -217,10 +217,7 @@ def _latex_(self):
         if self.is_zero():
             return "0"
         p = self.parent()
-        try:
-            names = p.latex_variable_names()
-        except ValueError:
-            names = None
+        names = p.latex_variable_names()
         if names:
             terms = [("T^{{({})}}{}".format(k1, names[p._index_to_pos[k0]]), v) if k1 > 1
                      else ("T{}".format(names[p._index_to_pos[k0]]), v) if k1 == 1
diff --git a/src/sage/groups/finitely_presented.py b/src/sage/groups/finitely_presented.py
index b65566ebeb4..1a4245692f0 100644
--- a/src/sage/groups/finitely_presented.py
+++ b/src/sage/groups/finitely_presented.py
@@ -751,10 +751,6 @@ def __init__(self, free_group, relations, category=None, libgap_fpgroup=None):
             <fp group of size 60 on the generators [ A_5.1, A_5.2 ]>
             sage: A5sage = A5gapfp.sage(); A5sage;
             Finitely presented group < A_5.1, A_5.2 | A_5.1^5*A_5.2^-5, A_5.1^5*(A_5.2^-1*A_5.1^-1)^2, (A_5.1^-2*A_5.2^2)^2 >
-            sage: A5sage.inject_variables()
-            Traceback (most recent call last):
-            ...
-            ValueError: variable names have not yet been set using self._assign_names(...)
 
         Check that pickling works::
 
@@ -772,10 +768,8 @@ def __init__(self, free_group, relations, category=None, libgap_fpgroup=None):
         assert isinstance(relations, tuple)
         self._free_group = free_group
         self._relations = relations
-        try:
+        if free_group.variable_names():
             self._assign_names(free_group.variable_names())
-        except ValueError:
-            pass
         if libgap_fpgroup is None:
             libgap_fpgroup = free_group.gap() / libgap([rel.gap() for rel in relations])
         ParentLibGAP.__init__(self, libgap_fpgroup)
diff --git a/src/sage/modules/free_module_element.pyx b/src/sage/modules/free_module_element.pyx
index 5826668750a..88a9ab045dc 100644
--- a/src/sage/modules/free_module_element.pyx
+++ b/src/sage/modules/free_module_element.pyx
@@ -3034,9 +3034,8 @@ cdef class FreeModuleElement(Vector):   # abstract base class
             [(x, y, z) |--> x, (x, y, z) |--> y, (x, y, z) |--> z]
         """
         R = self._parent.base_ring()
-        try:
-            var_names = R.variable_names()
-        except ValueError:
+        var_names = R.variable_names()
+        if not var_names:
             if hasattr(R, 'arguments'):
                 var_names = R.arguments()
             else:
diff --git a/src/sage/rings/asymptotic/misc.py b/src/sage/rings/asymptotic/misc.py
index 740529a374e..6c3b5e7795c 100644
--- a/src/sage/rings/asymptotic/misc.py
+++ b/src/sage/rings/asymptotic/misc.py
@@ -176,10 +176,13 @@ def abbreviate(P):
             op, cl = ('[', ']')
         else:
             op, cl = ('[[', ']]')
-        try:
-            s = abbreviate(P.base_ring()) + op + ', '.join(P.variable_names()) + cl
-        except ValueError:
-            s = str(P)
+        s = str(P)
+        if P.variable_names():
+            try:
+                s = abbreviate(P.base_ring()) + op + ', '.join(P.variable_names()) + cl
+            except ValueError:
+                pass
+
     else:
         try:
             s = abbreviate(P)
diff --git a/src/sage/rings/integer_ring.pyx b/src/sage/rings/integer_ring.pyx
index d5ca26fce5b..4a951464d4e 100644
--- a/src/sage/rings/integer_ring.pyx
+++ b/src/sage/rings/integer_ring.pyx
@@ -319,10 +319,14 @@ cdef class IntegerRing_class(CommutativeRing):
 
             sage: A in InfiniteEnumeratedSets()
             True
+
+            sage: ZZ.variable_names()
+            ()
+
         """
         cat = (EuclideanDomains(), DedekindDomains(),
                InfiniteEnumeratedSets().Metric(), NoetherianRings())
-        Parent.__init__(self, base=self, names=('x',), normalize=False,
+        Parent.__init__(self, base=self, normalize=False,
                         category=cat)
         self._populate_coercion_lists_(init_no_parent=True,
                                        convert_method_name='_integer_')
diff --git a/src/sage/rings/qqbar.py b/src/sage/rings/qqbar.py
index df010d70b84..0e4fa647f49 100644
--- a/src/sage/rings/qqbar.py
+++ b/src/sage/rings/qqbar.py
@@ -1075,40 +1075,6 @@ class AlgebraicRealField(Singleton, AlgebraicField_common, sage.rings.abc.Algebr
         True
     """
 
-    def __new__(cls):
-        r"""
-        This method is there to ensure that pickles created before this class
-        was made a :class:`~sage.misc.fast_methods.Singleton` still load.
-
-        TESTS::
-
-            sage: s = loads(b'x\x9cmQ\xcbR\x141\x14\xad\x11A\x083\xe2\x03T|'
-            ....: b'\x82l`\xd3\xff\xe0\x86\x8de/\xba*\xcb\xa9[\xe9\xf4'
-            ....: b'\xa5;e:=\'I+,\xa6J\x17B\xf9\xd7f\x08\xe2s\x95\xa4\xee9\xf7<'
-            ....: b'\xf2\xe5\x8e\x0e\xaa\xe5"D?\xea8z.\x9a\x0b\xa7z\xa3I[\x15'
-            ....: b'\x82\xf8\xf3\x85\xc9\xb1<xg[\xae\xbd2\xbabeO\r\xdb\x86>\x9b'
-            ....: b'\xd8\x91V\x91\xdb\xc1_\xe0f\xa57\xae\r\x05P+/\xfe\xe5\x08'
-            ....: b'\xaci\xa2z46\x1aG$Z\x8e*F/p\xf7oC\xa33\x18\x99</<\x07v\tf'
-            ....: b'\x06\'F\xe7\xb9\x195\x0b\xacg\xc2\x8d\xbc\xe1P\x9c\xad\x04'
-            ....: b'\x828\xcd\x076N\x96W\xb8WaSN\x17\xca\xa7\r9\r\xb6.+\x88Kl'
-            ....: b'\x97e\xb7\x16+LO\xbeb\xb6\xc4\xfdc)\x88\xfb\x9a\x9b&\x05'
-            ....: b'\xc0N)wI\x0f\xee\x13\xfbH=\xc7nh(U\xc2xP\xca\r\xd2\x8d'
-            ....: b'\x8a\n\x0fK\xb9\xf5+\xfe\xa3n3MV\x98\x80\xc7rr\xfe\r\xbbr'
-            ....: b'\x9bZv\xecU\x1c|\xc0\xde\x12O\xe4:\xd5*0\x9ev3\xb9C\x0b'
-            ....: b'\xa3?Z\xa6\xa4\x11R6<{?I\xa2l\xb9\xbf6;\xb8\\\xc6\xe0\xb1'
-            ....: b'\x9f\xb3\xf6&\xe8\xe2,\xb3R\x13\xf9\xf2\xe1\xda\x9c\xc0s'
-            ....: b'\xb9\xf7?.\xe1E7\xeb\xa6W\x15^&\x80q&\x1aeo\x93Y\x13"^\xcd'
-            ....: b'\xf1Z\xee\xdf\x92W\x18Z\xa4\xa6(\xd7\x867\xdf\x93\xad\x9fL'
-            ....: b'\xa5W\xff\x90\x89\x07s\x1c\xfe6\xd2\x03{\xcdy\xf4v\x8e\xa3'
-            ....: b'\xb1.~\x000\xc2\xe0\xa1')
-            sage: s is AA
-            True
-        """
-        try:
-            return AA
-        except BaseException:
-            return AlgebraicField_common.__new__(cls)
-
     def __init__(self):
         r"""
         Standard initialization function.
@@ -1126,9 +1092,15 @@ def __init__(self):
             True
             sage: AA.has_coerce_map_from(int)
             True
+
+        TESTS::
+
+            sage: AA.variable_names()
+            ()
+
         """
         from sage.categories.fields import Fields
-        AlgebraicField_common.__init__(self, self, ('x',), normalize=False, category=Fields().Infinite())
+        AlgebraicField_common.__init__(self, self, normalize=False, category=Fields().Infinite())
         self._populate_coercion_lists_([ZZ, QQ])
 
     def _element_constructor_(self, x):
diff --git a/src/sage/rings/quotient_ring.py b/src/sage/rings/quotient_ring.py
index dd892d4fd59..fe227135a63 100644
--- a/src/sage/rings/quotient_ring.py
+++ b/src/sage/rings/quotient_ring.py
@@ -296,10 +296,8 @@ def QuotientRing(R, I, names=None, **kwds):
         raise TypeError("R must be a ring")
     is_commutative = R in _CommRings
     if names is None:
-        try:
-            names = tuple([x + 'bar' for x in R.variable_names()])
-        except ValueError:  # no names are assigned
-            pass
+        if R.variable_names():
+            names = tuple(x + 'bar' for x in R.variable_names())
     else:
         names = normalize_names(R.ngens(), names)
     if kwds.get('implementation') == 'pbori':
@@ -549,12 +547,10 @@ def construction(self):
 
         # Is there a better generic way to distinguish between things like Z/pZ as a field and Z/pZ as a ring?
         from sage.rings.ring import Field
-        try:
-            names = self.variable_names()
-        except ValueError:
-            try:
-                names = self.cover_ring().variable_names()
-            except ValueError:
+        names = self.variable_names()
+        if not names:
+            names = self.cover_ring().variable_names()
+            if not names:
                 names = None
         if self in _CommRings:
             return QuotientFunctor(self.__I, names=names, domain=_CommRings,
diff --git a/src/sage/rings/quotient_ring_element.py b/src/sage/rings/quotient_ring_element.py
index c01d96d9b80..8e61912687e 100644
--- a/src/sage/rings/quotient_ring_element.py
+++ b/src/sage/rings/quotient_ring_element.py
@@ -220,11 +220,10 @@ def _repr_(self):
         # However, it may occur that no variable names are assigned.
         # That holds, in particular, if there are infinitely many
         # generators, as for Steenrod algebras.
-        try:
-            P.variable_names()
-        except ValueError:
-            return str(self.__rep)
-        with localvars(R, P.variable_names(), normalize=False):
+        if P.variable_names():
+            with localvars(R, P.variable_names(), normalize=False):
+                return str(self.__rep)
+        else:
             return str(self.__rep)
 
     def _latex_(self):
@@ -245,11 +244,10 @@ def _latex_(self):
         P = self.parent()
         R = P.cover_ring()
         # see _repr_ above for the idea
-        try:
-            P.variable_names()
-        except ValueError:
-            return self.__rep._latex_()
-        with localvars(R, P.variable_names(), normalize=False):
+        if P.variable_names():
+            with localvars(R, P.variable_names(), normalize=False):
+                return self.__rep._latex_()
+        else:
             return self.__rep._latex_()
 
     def __pari__(self):
diff --git a/src/sage/rings/rational_field.py b/src/sage/rings/rational_field.py
index 6337baf9f26..abda1366d6d 100644
--- a/src/sage/rings/rational_field.py
+++ b/src/sage/rings/rational_field.py
@@ -219,10 +219,10 @@ def __init__(self):
         TESTS::
 
             sage: TestSuite(QQ).run()
-            sage: QQ.variable_name()
-            'x'
+
             sage: QQ.variable_names()
-            ('x',)
+            ()
+
             sage: QQ._element_constructor_((2, 3))
             2/3
 
@@ -237,7 +237,6 @@ def __init__(self):
         Parent.__init__(self, base=self,
                         category=[QuotientFields().Metric(),
                                   NumberFields()])
-        self._assign_names(('x',), normalize=False)  # ?????
         self._populate_coercion_lists_(init_no_parent=True)
 
     _element_constructor_ = Rational
diff --git a/src/sage/structure/category_object.pyx b/src/sage/structure/category_object.pyx
index 8571811dd97..37fffdbaed0 100644
--- a/src/sage/structure/category_object.pyx
+++ b/src/sage/structure/category_object.pyx
@@ -462,9 +462,11 @@ cdef class CategoryObject(SageObject):
 
     def variable_names(self):
         """
-        Return the list of variable names corresponding to the generators.
+        Return the variable names corresponding to our generators.
 
-        OUTPUT: a tuple of strings
+        OUTPUT:
+
+        A (possibly empty) tuple of strings.
 
         EXAMPLES::
 
@@ -480,10 +482,16 @@ cdef class CategoryObject(SageObject):
             sage: T.<x> = InfinitePolynomialRing(ZZ)
             sage: T.variable_names()
             ('x',)
+
+        ::
+
+            sage: NN.variable_names()
+            ()
+
         """
         if self._names is not None:
             return self._names
-        raise ValueError("variable names have not yet been set using self._assign_names(...)")
+        return ()
 
     def variable_name(self):
         """
@@ -525,13 +533,11 @@ cdef class CategoryObject(SageObject):
         # We cannot assume that self *has* _latex_variable_names.
         # But there is a method that returns them and sets
         # the attribute at the same time, if needed.
-        # Simon King: It is not necessarily the case that variable
-        # names are assigned. In that case, self._names is None,
-        # and self.variable_names() raises a ValueError
-        try:
+
+        old = None, None
+        if self.variable_names():
             old = self.variable_names(), self.latex_variable_names()
-        except ValueError:
-            old = None, None
+
         self._names, self._latex_names = names, latex_names
         return old