diff --git a/src/sage/modules/all.py b/src/sage/modules/all.py
index 87621e61b29..660f1ef3807 100644
--- a/src/sage/modules/all.py
+++ b/src/sage/modules/all.py
@@ -1,4 +1,4 @@
-#*****************************************************************************
+# *****************************************************************************
 #       Copyright (C) 2005 William Stein <wstein@gmail.com>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
@@ -11,23 +11,30 @@
 #  The full text of the GPL is available at:
 #
 #                  http://www.gnu.org/licenses/
-#*****************************************************************************
+# *****************************************************************************
 
-from .free_module import FreeModule, VectorSpace, span
+from sage.modules.free_module import FreeModule, VectorSpace, span
 
-from .free_quadratic_module import (FreeQuadraticModule, QuadraticSpace,
-                                   InnerProductSpace)
+from sage.modules.free_quadratic_module import (
+    FreeQuadraticModule,
+    QuadraticSpace,
+    InnerProductSpace,
+)
 
-from .free_module_element import (vector, free_module_element, zero_vector,
-                                 random_vector)
+from sage.modules.free_module_element import (
+    vector,
+    free_module_element,
+    zero_vector,
+    random_vector,
+)
 
-from .vector_space_morphism import linear_transformation
+from sage.modules.vector_space_morphism import linear_transformation
 
-from .with_basis.all import *
+from sage.modules.with_basis.all import *
 
 from sage.misc.lazy_import import lazy_import
 
-lazy_import('sage.modules.filtered_vector_space', 'FilteredVectorSpace')
-lazy_import('sage.modules.multi_filtered_vector_space', 'MultiFilteredVectorSpace')
-lazy_import('sage.modules.free_quadratic_module_integer_symmetric', 'IntegralLattice')
-lazy_import('sage.modules.torsion_quadratic_module', 'TorsionQuadraticForm')
+lazy_import("sage.modules.filtered_vector_space", "FilteredVectorSpace")
+lazy_import("sage.modules.multi_filtered_vector_space", "MultiFilteredVectorSpace")
+lazy_import("sage.modules.free_quadratic_module_integer_symmetric", "IntegralLattice")
+lazy_import("sage.modules.torsion_quadratic_module", "TorsionQuadraticForm")
diff --git a/src/sage/modules/complex_double_vector.py b/src/sage/modules/complex_double_vector.py
index 88ad6bb6d67..728d862e8fc 100644
--- a/src/sage/modules/complex_double_vector.py
+++ b/src/sage/modules/complex_double_vector.py
@@ -19,6 +19,6 @@
 #                  http://www.gnu.org/licenses/
 ###############################################################################
 
-from .vector_complex_double_dense import Vector_complex_double_dense
+from sage.modules.vector_complex_double_dense import Vector_complex_double_dense
 
 ComplexDoubleVectorSpaceElement = Vector_complex_double_dense
diff --git a/src/sage/modules/free_module.py b/src/sage/modules/free_module.py
index 8f584a211ef..cadae277429 100644
--- a/src/sage/modules/free_module.py
+++ b/src/sage/modules/free_module.py
@@ -177,33 +177,41 @@
 ###########################################################################
 
 import itertools
+from warnings import warn
 
-from . import free_module_element
 import sage.matrix.matrix_space
 import sage.misc.latex as latex
-
-from sage.modules.module import Module
-import sage.rings.ring as ring
 import sage.rings.abc
-import sage.rings.integer_ring
-import sage.rings.rational_field
 import sage.rings.infinity
 import sage.rings.integer
-from sage.categories.principal_ideal_domains import PrincipalIdealDomains
-from sage.categories.integral_domains import IntegralDomains
+import sage.rings.integer_ring
+import sage.rings.rational_field
+import sage.rings.ring as ring
 from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
+from sage.categories.integral_domains import IntegralDomains
+from sage.categories.principal_ideal_domains import PrincipalIdealDomains
+from sage.misc.cachefunc import cached_method
 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.module import Module
 from sage.rings.finite_rings.finite_field_base import FiniteField
 from sage.structure.factory import UniqueFactory
+from sage.structure.richcmp import (
+    op_EQ,
+    op_GE,
+    op_GT,
+    op_LE,
+    op_LT,
+    op_NE,
+    revop,
+    rich_to_bool,
+    richcmp,
+    richcmp_method,
+    richcmp_not_equal,
+)
 from sage.structure.sequence import Sequence
-from sage.structure.richcmp import (richcmp_method, rich_to_bool, richcmp,
-                                    richcmp_not_equal, revop,
-                                    op_LT,op_LE,op_EQ,op_NE,op_GT,op_GE)
-from sage.misc.cachefunc import cached_method
-from sage.misc.lazy_import import LazyImport
-
-from warnings import warn
 
 ###############################################################################
 #
@@ -226,7 +234,7 @@ def create_key(self, base_ring, rank, sparse=False, inner_product_matrix=None):
         """
         rank = int(sage.rings.integer.Integer(rank))
 
-        if not (inner_product_matrix is None):
+        if inner_product_matrix is not None:
             inner_product_matrix = sage.matrix.matrix_space.MatrixSpace(base_ring, rank)(inner_product_matrix)
             inner_product_matrix.set_immutable()
 
@@ -1821,7 +1829,9 @@ def free_resolution(self, *args, **kwds):
                         [    z   x*z]
              0 <-- C_0 <-------------- C_1 <-- 0
         """
-        from sage.rings.polynomial.multi_polynomial_libsingular import MPolynomialRing_libsingular
+        from sage.rings.polynomial.multi_polynomial_libsingular import (
+            MPolynomialRing_libsingular,
+        )
         if isinstance(self.base_ring(), MPolynomialRing_libsingular):
             from sage.homology.free_resolution import FiniteFreeResolution_singular
             return FiniteFreeResolution_singular(self, *args, **kwds)
@@ -1854,13 +1864,19 @@ def graded_free_resolution(self, *args, **kwds):
             sage: N.graded_free_resolution(degrees=[2, 1, 3], shifts=[2, 3])            # needs sage.libs.singular
             S(-2)⊕S(-3) <-- S(-6)⊕S(-8) <-- 0
         """
-        from sage.rings.polynomial.multi_polynomial_libsingular import MPolynomialRing_libsingular
+        from sage.rings.polynomial.multi_polynomial_libsingular import (
+            MPolynomialRing_libsingular,
+        )
         if isinstance(self.base_ring(), MPolynomialRing_libsingular):
-            from sage.homology.graded_resolution import GradedFiniteFreeResolution_singular
+            from sage.homology.graded_resolution import (
+                GradedFiniteFreeResolution_singular,
+            )
             return GradedFiniteFreeResolution_singular(self, *args, **kwds)
 
         if isinstance(self, FreeModule_generic):
-            from sage.homology.graded_resolution import GradedFiniteFreeResolution_free_module
+            from sage.homology.graded_resolution import (
+                GradedFiniteFreeResolution_free_module,
+            )
             return GradedFiniteFreeResolution_free_module(self, *args, **kwds)
 
         raise NotImplementedError("the module must be a free module or "
@@ -4084,7 +4100,9 @@ def span_of_basis(self, basis, base_ring=None, check=True, already_echelonized=F
         if base_ring is None or base_ring == self.base_ring():
             try:
                 if self.is_dense():
-                    from .free_module_integer import FreeModule_submodule_with_basis_integer
+                    from .free_module_integer import (
+                        FreeModule_submodule_with_basis_integer,
+                    )
                     return FreeModule_submodule_with_basis_integer(self.ambient_module(),
                                                                    basis=basis, check=check,
                                                                    already_echelonized=already_echelonized,
@@ -5370,8 +5388,8 @@ def _coerce_map_from_(self, M):
             sage: V = QQ^2
             sage: V.coerce_map_from(M)
         """
-        from sage.modules.submodule import Submodule_free_ambient
         from sage.modules.quotient_module import FreeModule_ambient_field_quotient
+        from sage.modules.submodule import Submodule_free_ambient
 
         if isinstance(M, FreeModule_ambient_field_quotient):
             # No forgetful map.
@@ -5989,8 +6007,8 @@ def _sympy_(self):
             sage: (1, 2, 3) in sZZ3                                                     # needs sympy
             True
         """
-        from sympy import ProductSet
         from sage.interfaces.sympy import sympy_init
+        from sympy import ProductSet
         sympy_init()
         return ProductSet(*([self.coordinate_ring()] * self.rank()))
 
@@ -8194,7 +8212,7 @@ def element_class(R, is_sparse):
             else:
                 return Vector_mod2_dense
         try:
-            from .vector_modn_dense import Vector_modn_dense, MAX_MODULUS
+            from .vector_modn_dense import MAX_MODULUS, Vector_modn_dense
         except ImportError:
             pass
         else:
@@ -8210,7 +8228,9 @@ def element_class(R, is_sparse):
             return Vector_real_double_dense
     elif isinstance(R, sage.rings.abc.ComplexDoubleField) and not is_sparse:
         try:
-            from sage.modules.vector_complex_double_dense import Vector_complex_double_dense
+            from sage.modules.vector_complex_double_dense import (
+                Vector_complex_double_dense,
+            )
         except ImportError:
             pass
         else:
diff --git a/src/sage/modules/free_module_morphism.py b/src/sage/modules/free_module_morphism.py
index cdc933f23d0..6cf1add1f75 100644
--- a/src/sage/modules/free_module_morphism.py
+++ b/src/sage/modules/free_module_morphism.py
@@ -41,12 +41,10 @@
 # be coercible into vector space of appropriate dimension.
 
 import sage.modules.free_module as free_module
-from . import matrix_morphism
 from sage.categories.morphism import Morphism
+from sage.modules import free_module_homspace, matrix_morphism
+from sage.structure.richcmp import rich_to_bool, richcmp
 from sage.structure.sequence import Sequence
-from sage.structure.richcmp import richcmp, rich_to_bool
-
-from . import free_module_homspace
 
 
 def is_FreeModuleMorphism(x):
diff --git a/src/sage/modules/free_quadratic_module.py b/src/sage/modules/free_quadratic_module.py
index ff112b9a5c2..df0f90a4fb6 100644
--- a/src/sage/modules/free_quadratic_module.py
+++ b/src/sage/modules/free_quadratic_module.py
@@ -71,7 +71,7 @@
 import sage.misc.latex as latex
 import sage.rings.ring as ring
 from sage.categories.principal_ideal_domains import PrincipalIdealDomains
-from . import free_module
+from sage.modules import free_module
 
 # #############################################################################
 #
@@ -154,7 +154,7 @@ def FreeQuadraticModule(base_ring, rank, inner_product_matrix,
 
     if key in _cache:
         M = _cache[key]()
-        if not (M is None):
+        if M is not None:
             return M
 
     if not base_ring.is_commutative():
diff --git a/src/sage/modules/quotient_module.py b/src/sage/modules/quotient_module.py
index c50c1d044fc..09c8b313c91 100644
--- a/src/sage/modules/quotient_module.py
+++ b/src/sage/modules/quotient_module.py
@@ -19,10 +19,11 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from .free_module import (Module_free_ambient,
-                          FreeModule_ambient,
-                          FreeModule_ambient_field)
-
+from sage.modules.free_module import (
+    FreeModule_ambient,
+    FreeModule_ambient_field,
+    Module_free_ambient,
+)
 
 ###############################################################################
 #
diff --git a/src/sage/modules/real_double_vector.py b/src/sage/modules/real_double_vector.py
index 1ceb320078e..5430e175f11 100644
--- a/src/sage/modules/real_double_vector.py
+++ b/src/sage/modules/real_double_vector.py
@@ -16,6 +16,6 @@
 #  Distributed under the terms of the GNU General Public License (GPL)
 #                  http://www.gnu.org/licenses/
 ###############################################################################
-from .vector_real_double_dense import Vector_real_double_dense
+from sage.modules.vector_real_double_dense import Vector_real_double_dense
 
 RealDoubleVectorSpaceElement = Vector_real_double_dense
diff --git a/src/sage/modules/submodule.py b/src/sage/modules/submodule.py
index 3ab5a195f49..8d00d4f4553 100644
--- a/src/sage/modules/submodule.py
+++ b/src/sage/modules/submodule.py
@@ -34,10 +34,12 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from sage.modules.free_module import (basis_seq,
-                                      Module_free_ambient,
-                                      FreeModule_ambient_domain)
-from .quotient_module import QuotientModule_free_ambient
+from sage.modules.free_module import (
+    FreeModule_ambient_domain,
+    Module_free_ambient,
+    basis_seq,
+)
+from sage.modules.quotient_module import QuotientModule_free_ambient
 
 
 class Submodule_free_ambient(Module_free_ambient):
diff --git a/src/sage/modules/vector_callable_symbolic_dense.py b/src/sage/modules/vector_callable_symbolic_dense.py
index 1929eabcf46..70b05d98b66 100644
--- a/src/sage/modules/vector_callable_symbolic_dense.py
+++ b/src/sage/modules/vector_callable_symbolic_dense.py
@@ -49,7 +49,7 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from . import free_module_element
+from sage.modules import free_module_element
 from sage.symbolic.ring import SR
 
 
diff --git a/src/sage/modules/vector_complex_double_dense.pxd b/src/sage/modules/vector_complex_double_dense.pxd
index f26526a1669..1681bad8c34 100644
--- a/src/sage/modules/vector_complex_double_dense.pxd
+++ b/src/sage/modules/vector_complex_double_dense.pxd
@@ -1,4 +1,4 @@
-from .vector_double_dense cimport Vector_double_dense
+from sage.modules .vector_double_dense cimport Vector_double_dense
 
 cdef class Vector_complex_double_dense(Vector_double_dense):
     pass
diff --git a/src/sage/modules/vector_double_dense.pxd b/src/sage/modules/vector_double_dense.pxd
index f29b4597d15..c378db6c1d6 100644
--- a/src/sage/modules/vector_double_dense.pxd
+++ b/src/sage/modules/vector_double_dense.pxd
@@ -1,4 +1,4 @@
-from .vector_numpy_dense cimport Vector_numpy_dense
+from sage.modules .vector_numpy_dense cimport Vector_numpy_dense
 
 cdef class Vector_double_dense(Vector_numpy_dense):
     pass
diff --git a/src/sage/modules/vector_integer_dense.pxd b/src/sage/modules/vector_integer_dense.pxd
index 6a221546b34..ebd69249b70 100644
--- a/src/sage/modules/vector_integer_dense.pxd
+++ b/src/sage/modules/vector_integer_dense.pxd
@@ -1,4 +1,4 @@
-from .free_module_element cimport FreeModuleElement
+from sage.modules .free_module_element cimport FreeModuleElement
 from sage.libs.gmp.types cimport mpz_t
 from sage.structure.parent cimport Parent
 
diff --git a/src/sage/modules/vector_mod2_dense.pxd b/src/sage/modules/vector_mod2_dense.pxd
index a5542fe1345..06dc10fc841 100644
--- a/src/sage/modules/vector_mod2_dense.pxd
+++ b/src/sage/modules/vector_mod2_dense.pxd
@@ -1,4 +1,4 @@
-from .free_module_element cimport FreeModuleElement
+from sage.modules .free_module_element cimport FreeModuleElement
 from sage.libs.m4ri cimport mzd_t
 
 
diff --git a/src/sage/modules/vector_modn_dense.pxd b/src/sage/modules/vector_modn_dense.pxd
index 4e0aff593fe..b3a7df6aac7 100644
--- a/src/sage/modules/vector_modn_dense.pxd
+++ b/src/sage/modules/vector_modn_dense.pxd
@@ -1,4 +1,4 @@
-from .free_module_element cimport FreeModuleElement
+from sage.modules .free_module_element cimport FreeModuleElement
 from sage.ext.mod_int cimport mod_int
 
 cdef class Vector_modn_dense(FreeModuleElement):
diff --git a/src/sage/modules/vector_numpy_dense.pxd b/src/sage/modules/vector_numpy_dense.pxd
index b019bc8ebac..ec1bea61727 100644
--- a/src/sage/modules/vector_numpy_dense.pxd
+++ b/src/sage/modules/vector_numpy_dense.pxd
@@ -1,4 +1,4 @@
-from .free_module_element cimport FreeModuleElement
+from sage.modules .free_module_element cimport FreeModuleElement
 cimport numpy
 
 cdef class Vector_numpy_dense(FreeModuleElement):
diff --git a/src/sage/modules/vector_numpy_dense.pyx b/src/sage/modules/vector_numpy_dense.pyx
index f0e2224b96d..8e0650aa90f 100644
--- a/src/sage/modules/vector_numpy_dense.pyx
+++ b/src/sage/modules/vector_numpy_dense.pyx
@@ -29,7 +29,7 @@ AUTHORS:
 
 cimport numpy
 import numpy
-from .free_module_element import FreeModuleElement
+from sage.modules .free_module_element import FreeModuleElement
 
 # This is for the NumPy C API (the PyArray... functions) to work
 numpy.import_array()
diff --git a/src/sage/modules/vector_numpy_integer_dense.pxd b/src/sage/modules/vector_numpy_integer_dense.pxd
index e39b90bbcb9..e2e3dbf8359 100644
--- a/src/sage/modules/vector_numpy_integer_dense.pxd
+++ b/src/sage/modules/vector_numpy_integer_dense.pxd
@@ -1,4 +1,4 @@
-from .vector_numpy_dense cimport Vector_numpy_dense
+from sage.modules .vector_numpy_dense cimport Vector_numpy_dense
 
 cdef class Vector_numpy_integer_dense(Vector_numpy_dense):
 
diff --git a/src/sage/modules/vector_rational_dense.pxd b/src/sage/modules/vector_rational_dense.pxd
index 34db6f6252b..b70048c8705 100644
--- a/src/sage/modules/vector_rational_dense.pxd
+++ b/src/sage/modules/vector_rational_dense.pxd
@@ -1,4 +1,4 @@
-from .free_module_element cimport FreeModuleElement
+from sage.modules .free_module_element cimport FreeModuleElement
 from sage.libs.gmp.types cimport mpq_t
 from sage.structure.parent cimport Parent
 
diff --git a/src/sage/modules/vector_real_double_dense.pxd b/src/sage/modules/vector_real_double_dense.pxd
index 6f75ec0c4dd..e5d4ad284b5 100644
--- a/src/sage/modules/vector_real_double_dense.pxd
+++ b/src/sage/modules/vector_real_double_dense.pxd
@@ -1,4 +1,4 @@
-from .vector_double_dense cimport Vector_double_dense
+from sage.modules .vector_double_dense cimport Vector_double_dense
 
 cdef class Vector_real_double_dense(Vector_double_dense):
     pass
diff --git a/src/sage/modules/vector_space_morphism.py b/src/sage/modules/vector_space_morphism.py
index 2685462a74e..97c99d8da5b 100644
--- a/src/sage/modules/vector_space_morphism.py
+++ b/src/sage/modules/vector_space_morphism.py
@@ -329,11 +329,12 @@
 ####################################################################################
 
 
-import sage.modules.matrix_morphism as matrix_morphism
 import sage.modules.free_module_morphism as free_module_morphism
-from . import vector_space_homspace
+import sage.modules.matrix_morphism as matrix_morphism
+from sage.modules import vector_space_homspace
 from sage.structure.element import is_Matrix
 
+
 def linear_transformation(arg0, arg1=None, arg2=None, side='left'):
     r"""
     Create a linear transformation from a variety of possible inputs.
@@ -692,12 +693,14 @@ def linear_transformation(arg0, arg1=None, arg2=None, side='left'):
         ArithmeticError: some image of the function is not in the codomain, because
         element [1, 0] is not in free module
     """
+    from sage.categories.homset import Hom
     from sage.matrix.constructor import matrix
-    from sage.modules.module import is_VectorSpace
     from sage.modules.free_module import VectorSpace
-    from sage.categories.homset import Hom
+    from sage.modules.module import is_VectorSpace
     try:
-        from sage.modules.vector_callable_symbolic_dense import Vector_callable_symbolic_dense
+        from sage.modules.vector_callable_symbolic_dense import (
+            Vector_callable_symbolic_dense,
+        )
     except ImportError:
         Vector_callable_symbolic_dense = ()
 
diff --git a/src/sage/modules/vector_symbolic_dense.py b/src/sage/modules/vector_symbolic_dense.py
index ce134c0462e..46ea21be585 100644
--- a/src/sage/modules/vector_symbolic_dense.py
+++ b/src/sage/modules/vector_symbolic_dense.py
@@ -54,7 +54,7 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from . import free_module_element
+from sage.modules import free_module_element
 from sage.symbolic.expression import Expression
 
 
diff --git a/src/sage/modules/vector_symbolic_sparse.py b/src/sage/modules/vector_symbolic_sparse.py
index 73cb41ae611..ff53995021c 100644
--- a/src/sage/modules/vector_symbolic_sparse.py
+++ b/src/sage/modules/vector_symbolic_sparse.py
@@ -56,7 +56,7 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from . import free_module_element
+from sage.modules import free_module_element
 from sage.symbolic.expression import Expression