diff --git a/src/sage/categories/category_singleton.pyx b/src/sage/categories/category_singleton.pyx
index 0423ab9d83a..83f61842034 100644
--- a/src/sage/categories/category_singleton.pyx
+++ b/src/sage/categories/category_singleton.pyx
@@ -10,7 +10,6 @@ Singleton categories
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from sage.misc.constant_function import ConstantFunction
 from sage.misc.lazy_attribute import lazy_class_attribute
 from sage.categories.category import Category
 from sage.structure.category_object cimport CategoryObject
@@ -318,10 +317,12 @@ class Category_singleton(Category):
             ...
             AssertionError: <class '__main__.MySubStuff'> is not a direct subclass of <class 'sage.categories.category_singleton.Category_singleton'>
         """
+        from sage.misc.constant_function import ConstantFunction
+        from sage.categories.category_with_axiom import CategoryWithAxiom_singleton
+
         if isinstance(cls, DynamicMetaclass):  # cls is something like Rings_with_category
             cls = cls.__base__
         # TODO: find a better way to check that cls is an abstract class
-        from sage.categories.category_with_axiom import CategoryWithAxiom_singleton
         assert (cls.__mro__[1] is Category_singleton or cls.__mro__[1] is CategoryWithAxiom_singleton), \
             "{} is not a direct subclass of {}".format(cls, Category_singleton)
         obj = super().__classcall__(cls, *args)
diff --git a/src/sage/categories/morphism.pyx b/src/sage/categories/morphism.pyx
index 28e89a36299..19b705b1721 100644
--- a/src/sage/categories/morphism.pyx
+++ b/src/sage/categories/morphism.pyx
@@ -39,8 +39,6 @@ AUTHORS:
 
 from cpython.object cimport *
 
-from sage.misc.constant_function import ConstantFunction
-
 from sage.structure.element cimport Element, ModuleElement
 from sage.structure.richcmp cimport richcmp_not_equal, rich_to_bool
 from sage.structure.parent cimport Parent
@@ -92,6 +90,8 @@ cdef class Morphism(Map):
             sage: phi
             Defunct morphism
         """
+        from sage.misc.constant_function import ConstantFunction
+
         D = self.domain()
         if D is None:
             return "Defunct morphism"
diff --git a/src/sage/manifolds/differentiable/characteristic_cohomology_class.py b/src/sage/manifolds/differentiable/characteristic_cohomology_class.py
index 63d0384e149..fd10ab41491 100644
--- a/src/sage/manifolds/differentiable/characteristic_cohomology_class.py
+++ b/src/sage/manifolds/differentiable/characteristic_cohomology_class.py
@@ -281,13 +281,13 @@
 # *****************************************************************************
 
 from sage.algebras.finite_gca import FiniteGCAlgebra
-from sage.combinat.free_module import IndexedFreeModuleElement
 from sage.manifolds.differentiable.affine_connection import AffineConnection
 from sage.manifolds.differentiable.bundle_connection import BundleConnection
 from sage.manifolds.differentiable.levi_civita_connection import LeviCivitaConnection
 from sage.misc.abstract_method import abstract_method
 from sage.misc.cachefunc import cached_method
 from sage.misc.fast_methods import Singleton
+from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
 from sage.rings.polynomial.polynomial_element import Polynomial
 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.structure.sage_object import SageObject
diff --git a/src/sage/misc/lazy_attribute.pyi b/src/sage/misc/lazy_attribute.pyi
index 6a174a37aa2..b857216eed1 100644
--- a/src/sage/misc/lazy_attribute.pyi
+++ b/src/sage/misc/lazy_attribute.pyi
@@ -1,5 +1,8 @@
 # This type-stub file helps pyright understand the decorator @lazy_attribute.
 
+from collections.abc import Callable
+from typing import Any
+
 # Adapted from https://github.com/python/typeshed/blob/b9640005eb586afdbe0a57bac2b88a7a12465069/stdlib/builtins.pyi#L1237-L1254
 class lazy_attribute:
     def __init__(
diff --git a/src/sage/modules/free_module.py b/src/sage/modules/free_module.py
index ba8ac43f52c..2e94cf1e608 100644
--- a/src/sage/modules/free_module.py
+++ b/src/sage/modules/free_module.py
@@ -194,7 +194,11 @@
 from sage.misc.lazy_attribute import lazy_attribute
 from sage.misc.lazy_import import LazyImport
 from sage.misc.randstate import current_randstate
-from sage.modules import free_module_element
+from sage.modules.free_module_element import (
+    FreeModuleElement,
+    FreeModuleElement_generic_dense,
+    FreeModuleElement_generic_sparse,
+)
 from sage.modules.module import Module
 from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.structure.factory import UniqueFactory
@@ -213,13 +217,13 @@
 )
 from sage.structure.sequence import Sequence
 
-
 ###############################################################################
 #
 # Constructor functions
 #
 ###############################################################################
 
+
 class FreeModuleFactory(UniqueFactory):
     r"""
     Factory class for the finite-dimensional free modules with standard basis
@@ -776,7 +780,7 @@ def span(gens, base_ring=None, check=True, already_echelonized=False):
         return FreeModule(R, 0)
     else:
         x = gens[0]
-        if isinstance(x, free_module_element.FreeModuleElement):
+        if isinstance(x, FreeModuleElement):
             M = x.parent()
         else:
             try:
@@ -950,7 +954,7 @@ def _element_constructor_(self, x, coerce=True, copy=True, check=True):
         """
         if isinstance(x, (int, sage.rings.integer.Integer)) and x == 0:
             return self.zero_vector()
-        elif isinstance(x, free_module_element.FreeModuleElement):
+        elif isinstance(x, FreeModuleElement):
             if x.parent() is self:
                 if copy:
                     return x.__copy__()
@@ -2223,7 +2227,7 @@ def _element_constructor_(self, x, coerce=True, copy=True, check=True):
         """
         if (isinstance(x, (int, sage.rings.integer.Integer)) and x == 0):
             return self.zero_vector()
-        if isinstance(x, free_module_element.FreeModuleElement):
+        if isinstance(x, FreeModuleElement):
             if x.parent() is self:
                 if copy:
                     return x.__copy__()
@@ -7040,7 +7044,7 @@ def echelon_coordinates(self, v, check=True):
             sage: W.echelon_coordinates([0,0,2,0,-1/2])
             [0, 2]
         """
-        if not isinstance(v, free_module_element.FreeModuleElement):
+        if not isinstance(v, FreeModuleElement):
             v = self.ambient_vector_space()(v)
         elif v.degree() != self.degree():
             raise ArithmeticError("vector is not in free module")
@@ -8069,7 +8073,7 @@ def echelon_coordinates(self, v, check=True):
             sage: vector(QQ, W.echelon_coordinates(v)) * W.basis_matrix()
             (1, 5, 9)
         """
-        if not isinstance(v, free_module_element.FreeModuleElement):
+        if not isinstance(v, FreeModuleElement):
             v = self.ambient_vector_space()(v)
         if v.degree() != self.degree():
             raise ArithmeticError("v (=%s) is not in self" % v)
@@ -8213,7 +8217,7 @@ def element_class(R, is_sparse):
         else:
             if R.order() < MAX_MODULUS:
                 return Vector_modn_dense
-        return free_module_element.FreeModuleElement_generic_dense
+        return FreeModuleElement_generic_dense
     elif isinstance(R, sage.rings.abc.RealDoubleField) and not is_sparse:
         try:
             from sage.modules.vector_real_double_dense import Vector_real_double_dense
@@ -8242,9 +8246,9 @@ def element_class(R, is_sparse):
             return sage.modules.vector_symbolic_sparse.Vector_symbolic_sparse
 
     if is_sparse:
-        return free_module_element.FreeModuleElement_generic_sparse
+        return FreeModuleElement_generic_sparse
     else:
-        return free_module_element.FreeModuleElement_generic_dense
+        return FreeModuleElement_generic_dense
 
 
 @richcmp_method
diff --git a/src/sage/rings/integer_ring.pxd b/src/sage/rings/integer_ring.pxd
index 204ccbe141c..fb7e0d97b5b 100644
--- a/src/sage/rings/integer_ring.pxd
+++ b/src/sage/rings/integer_ring.pxd
@@ -1,5 +1,4 @@
 from sage.rings.ring cimport CommutativeRing
-from sage.rings.integer cimport Integer
 from sage.libs.gmp.types cimport mpz_t
 
 cdef class IntegerRing_class(CommutativeRing):
diff --git a/src/sage/rings/integer_ring.pyx b/src/sage/rings/integer_ring.pyx
index d6555ac0eab..9d8bc4f09bd 100644
--- a/src/sage/rings/integer_ring.pyx
+++ b/src/sage/rings/integer_ring.pyx
@@ -66,9 +66,7 @@ from sage.structure.richcmp cimport rich_to_bool
 from sage.misc.misc_c import prod
 from sage.misc.randstate cimport randstate, current_randstate, SAGE_RAND_MAX
 
-cimport sage.rings.integer as integer
-
-from sage.rings import ring
+from sage.rings.integer cimport Integer
 
 arith = None
 cdef void late_import() noexcept:
@@ -329,7 +327,7 @@ cdef class IntegerRing_class(CommutativeRing):
         self._populate_coercion_lists_(init_no_parent=True,
                                        convert_method_name='_integer_')
 
-    _element_constructor_ = integer.Integer
+    _element_constructor_ = Integer
 
     def __reduce__(self):
         """
@@ -479,20 +477,20 @@ cdef class IntegerRing_class(CommutativeRing):
         if step is None:
             step = 1
         if type(step) is not int:
-            if not isinstance(step, integer.Integer):
-                step = integer.Integer(step)
+            if not isinstance(step, Integer):
+                step = Integer(step)
             if mpz_fits_slong_p((<Integer>step).value):
                 step = int(step)
-        if not isinstance(start, integer.Integer):
-            start = integer.Integer(start)
-        if not isinstance(end, integer.Integer):
-            end = integer.Integer(end)
-        cdef integer.Integer a = <Integer>start
-        cdef integer.Integer b = <Integer>end
+        if not isinstance(start, Integer):
+            start = Integer(start)
+        if not isinstance(end, Integer):
+            end = Integer(end)
+        cdef Integer a = <Integer>start
+        cdef Integer b = <Integer>end
 
         cdef int step_sign
         cdef long istep
-        cdef integer.Integer zstep, last
+        cdef Integer zstep, last
 
         L = []
         if type(step) is int:
@@ -797,8 +795,8 @@ cdef class IntegerRing_class(CommutativeRing):
             sage: ZZ.random_element() # indirect doctest # random
             6
         """
-        cdef integer.Integer r
-        cdef integer.Integer n_max, n_min, n_width
+        cdef Integer r
+        cdef Integer n_max, n_min, n_width
         cdef randstate rstate = current_randstate()
         cdef int den = rstate.c_random()-SAGE_RAND_MAX/2
         if den == 0: den = 1
@@ -809,11 +807,11 @@ cdef class IntegerRing_class(CommutativeRing):
                 if x is None:
                     mpz_set_si(value, rstate.c_random()%5 - 2)
                 else:
-                    n_max = x if isinstance(x, integer.Integer) else self(x)
+                    n_max = x if isinstance(x, Integer) else self(x)
                     mpz_urandomm(value, rstate.gmp_state, n_max.value)
             else:
-                n_min = x if isinstance(x, integer.Integer) else self(x)
-                n_max = y if isinstance(y, integer.Integer) else self(y)
+                n_min = x if isinstance(x, Integer) else self(x)
+                n_max = y if isinstance(y, Integer) else self(y)
                 n_width = n_max - n_min
                 if mpz_sgn(n_width.value) <= 0:
                     n_min = self(-2)
@@ -954,7 +952,7 @@ cdef class IntegerRing_class(CommutativeRing):
             ...
             TypeError: I must be an ideal of ZZ
         """
-        if isinstance(I, sage.rings.integer.Integer):
+        if isinstance(I, Integer):
             n = I
         elif isinstance(I, sage.rings.ideal.Ideal_generic):
             if not (I.ring() is self):
@@ -1023,7 +1021,7 @@ cdef class IntegerRing_class(CommutativeRing):
             ...
             TypeError: 96 is not prime
         """
-        if isinstance(prime, sage.rings.integer.Integer):
+        if isinstance(prime, Integer):
             p = self.ideal(prime)
         elif isinstance(prime, sage.rings.ideal.Ideal_generic):
             if not (prime.ring() is self):
@@ -1211,9 +1209,9 @@ cdef class IntegerRing_class(CommutativeRing):
             ValueError: n must be positive in zeta()
         """
         if n == 1:
-            return sage.rings.integer.Integer(1)
+            return Integer(1)
         elif n == 2:
-            return sage.rings.integer.Integer(-1)
+            return Integer(-1)
         elif n < 1:
             raise ValueError("n must be positive in zeta()")
         else:
diff --git a/src/sage/rings/rational.pyx b/src/sage/rings/rational.pyx
index 68ee004a251..0ee311f273b 100644
--- a/src/sage/rings/rational.pyx
+++ b/src/sage/rings/rational.pyx
@@ -65,7 +65,7 @@ from cysignals.signals cimport sig_on, sig_off
 import operator
 import fractions
 
-import sage.rings.rational_field
+from sage.rings.rational_field import QQ
 
 from sage.arith.long cimport integer_check_long_py
 from sage.categories.morphism cimport Morphism
@@ -73,7 +73,6 @@ from sage.categories.map cimport Map
 from sage.cpython.string cimport char_to_str, str_to_bytes
 from sage.libs.gmp.pylong cimport mpz_set_pylong
 from sage.rings.integer cimport Integer, smallInteger
-from sage.rings.integer_ring import ZZ
 from sage.structure.coerce cimport coercion_model, is_numpy_type
 from sage.structure.element cimport Element
 from sage.structure.parent cimport Parent
@@ -182,7 +181,7 @@ cdef Rational_sub_(Rational self, Rational other):
 
     return x
 
-cdef Parent the_rational_ring = sage.rings.rational_field.Q
+cdef Parent the_rational_ring = QQ
 
 # make sure zero/one elements are set
 cdef set_zero_one_elements():
@@ -2105,10 +2104,14 @@ cdef class Rational(sage.structure.element.FieldElement):
 
         num, exact = self.numerator().nth_root(n, 1)
         if not exact:
+            from sage.rings.integer_ring import ZZ
+
             raise ValueError("not a perfect %s power" % ZZ(n).ordinal_str())
 
         den, exact = self.denominator().nth_root(n, 1)
         if not exact:
+            from sage.rings.integer_ring import ZZ
+
             raise ValueError("not a perfect %s power" % ZZ(n).ordinal_str())
 
         if negative:
@@ -3195,6 +3198,8 @@ cdef class Rational(sage.structure.element.FieldElement):
             sage: (-1/2).log(3)                                                         # needs sage.symbolic
             (I*pi + log(1/2))/log(3)
         """
+        from sage.rings.integer_ring import ZZ
+
         cdef int self_sgn
         if self.denom().is_one():
             return ZZ(self.numer()).log(m, prec)
@@ -4119,10 +4124,10 @@ cdef class Z_to_Q(Morphism):
               From: Integer Ring
               To:   Rational Field
         """
-        from sage.rings import integer_ring
-        from sage.rings import rational_field
+        from sage.rings.integer_ring import ZZ
+
         import sage.categories.homset
-        Morphism.__init__(self, sage.categories.homset.Hom(integer_ring.ZZ, rational_field.QQ))
+        Morphism.__init__(self, sage.categories.homset.Hom(ZZ, QQ))
 
     cpdef Element _call_(self, x):
         """
diff --git a/src/sage/structure/sequence.py b/src/sage/structure/sequence.py
index 614ad8c0fe4..4ba6fea120f 100644
--- a/src/sage/structure/sequence.py
+++ b/src/sage/structure/sequence.py
@@ -71,8 +71,7 @@
 # ****************************************************************************
 
 from sage.misc.persist import register_unpickle_override
-import sage.structure.sage_object
-import sage.structure.coerce
+from sage.structure.sage_object import SageObject
 
 
 def Sequence(x, universe=None, check=True, immutable=False, cr=False, cr_str=None, use_sage_types=False):
@@ -306,7 +305,7 @@ def Sequence(x, universe=None, check=True, immutable=False, cr=False, cr_str=Non
     return Sequence_generic(x, universe, check, immutable, cr, cr_str, use_sage_types)
 
 
-class Sequence_generic(sage.structure.sage_object.SageObject, list):
+class Sequence_generic(SageObject, list):
     """
     A mutable list of elements with a common guaranteed universe,
     which can be set immutable.