diff --git a/src/sage/rings/abc.pxd b/src/sage/rings/abc.pxd
index a53b512d62f..a2d2e3f0ce4 100644
--- a/src/sage/rings/abc.pxd
+++ b/src/sage/rings/abc.pxd
@@ -1,4 +1,4 @@
-from .ring cimport CommutativeRing, Field
+from sage.rings.ring cimport CommutativeRing, Field
 
 cdef class RealField(Field):
 
diff --git a/src/sage/rings/all.py b/src/sage/rings/all.py
index 1d7b89601f5..2c2f6a0523e 100644
--- a/src/sage/rings/all.py
+++ b/src/sage/rings/all.py
@@ -12,10 +12,10 @@
 # ****************************************************************************
 from sage.misc.lazy_import import lazy_import
 
-from .all__sagemath_categories import *
+from sage.rings.all__sagemath_categories import *
 
 # Ring base classes
-from .ring import (Ring, Field, CommutativeRing, IntegralDomain,
+from sage.rings.ring import (Ring, Field, CommutativeRing, IntegralDomain,
     DedekindDomain, PrincipalIdealDomain, EuclideanDomain)
 
 # Ring element base classes
@@ -25,22 +25,22 @@
         EuclideanDomainElement, FieldElement)
 
 # Ideals
-from .ideal import Ideal
+from sage.rings.ideal import Ideal
 ideal = Ideal
 
 # Quotient
-from .quotient_ring import QuotientRing
+from sage.rings.quotient_ring import QuotientRing
 
 # Infinities
-from .infinity import infinity, Infinity, InfinityRing, unsigned_infinity, UnsignedInfinityRing
+from sage.rings.infinity import infinity, Infinity, InfinityRing, unsigned_infinity, UnsignedInfinityRing
 
 # Rational integers.
-from .integer_ring import IntegerRing, ZZ, crt_basis
-from .integer import Integer
+from sage.rings.integer_ring import IntegerRing, ZZ, crt_basis
+from sage.rings.integer import Integer
 
 # Rational numbers
-from .rational_field import RationalField, QQ
-from .rational import Rational
+from sage.rings.rational_field import RationalField, QQ
+from sage.rings.rational import Rational
 Rationals = RationalField
 
 # Integers modulo n.
@@ -49,120 +49,120 @@
 Integers = IntegerModRing
 
 # Finite fields
-from .finite_rings.all import *
+from sage.rings.finite_rings.all import *
 
 # Number field
-from .number_field.all import *
+from sage.rings.number_field.all import *
 
 # Function field
-from .function_field.all import *
+from sage.rings.function_field.all import *
 
 # Finite residue fields
-from .finite_rings.residue_field import ResidueField
+from sage.rings.finite_rings.residue_field import ResidueField
 
 # p-adic field
-from .padics.all import *
-from .padics.padic_printing import _printer_defaults as padic_printing
+from sage.rings.padics.all import *
+from sage.rings.padics.padic_printing import _printer_defaults as padic_printing
 
 # valuations
-from .valuation.all import *
+from sage.rings.valuation.all import *
 
 # Semirings
-from .semirings.all import *
+from sage.rings.semirings.all import *
 
 # Real numbers
-from .real_mpfr import (RealField, RR,
+from sage.rings.real_mpfr import (RealField, RR,
                        create_RealNumber as RealNumber)   # this is used by the preparser to wrap real literals -- very important.
 Reals = RealField
 
-from .real_double import RealDoubleField, RDF, RealDoubleElement
+from sage.rings.real_double import RealDoubleField, RDF, RealDoubleElement
 
-from .real_lazy import RealLazyField, RLF, ComplexLazyField, CLF
+from sage.rings.real_lazy import RealLazyField, RLF, ComplexLazyField, CLF
 
 from sage.rings.real_arb import RealBallField, RBF
 
 # Polynomial Rings and Polynomial Quotient Rings
-from .polynomial.all import *
+from sage.rings.polynomial.all import *
 
 
 # Algebraic numbers
-from .qqbar import (AlgebraicRealField, AA,
+from sage.rings.qqbar import (AlgebraicRealField, AA,
                    AlgebraicReal,
                    AlgebraicField, QQbar,
                    AlgebraicNumber,
                    number_field_elements_from_algebraics)
-from .universal_cyclotomic_field import UniversalCyclotomicField, E
+from sage.rings.universal_cyclotomic_field import UniversalCyclotomicField, E
 
 # Intervals
-from .real_mpfi import (RealIntervalField,
+from sage.rings.real_mpfi import (RealIntervalField,
                        RIF,
                        RealInterval)
 
 # Complex numbers
-from .complex_mpfr import ComplexField
-from .complex_mpfr import create_ComplexNumber as ComplexNumber
+from sage.rings.complex_mpfr import ComplexField
+from sage.rings.complex_mpfr import create_ComplexNumber as ComplexNumber
 Complexes = ComplexField
-from .complex_interval_field import ComplexIntervalField
-from .complex_interval import (create_ComplexIntervalFieldElement as ComplexIntervalFieldElement)
+from sage.rings.complex_interval_field import ComplexIntervalField
+from sage.rings.complex_interval import (create_ComplexIntervalFieldElement as ComplexIntervalFieldElement)
 
-from .complex_double import ComplexDoubleField, ComplexDoubleElement, CDF
+from sage.rings.complex_double import ComplexDoubleField, ComplexDoubleElement, CDF
 
-from .complex_mpc import MPComplexField
+from sage.rings.complex_mpc import MPComplexField
 
 from sage.rings.complex_arb import ComplexBallField, CBF
 
 lazy_import("sage.rings.imaginary_unit", "I")
 
 # Power series rings
-from .power_series_ring import PowerSeriesRing
+from sage.rings.power_series_ring import PowerSeriesRing
 
 # Laurent series ring in one variable
-from .laurent_series_ring import LaurentSeriesRing
+from sage.rings.laurent_series_ring import LaurentSeriesRing
 
 # Lazy Laurent series ring
 lazy_import('sage.rings.lazy_series_ring', ['LazyLaurentSeriesRing', 'LazyPowerSeriesRing',
                                             'LazySymmetricFunctions', 'LazyDirichletSeriesRing'])
 
 # Tate algebras
-from .tate_algebra import TateAlgebra
+from sage.rings.tate_algebra import TateAlgebra
 
 # Puiseux series ring
-from .puiseux_series_ring import PuiseuxSeriesRing
+from sage.rings.puiseux_series_ring import PuiseuxSeriesRing
 
 # Pseudo-ring of PARI objects.
-from .pari_ring import PariRing, Pari
+from sage.rings.pari_ring import PariRing, Pari
 
 # Big-oh notation
-from .big_oh import O
+from sage.rings.big_oh import O
 
 # Fraction field
-from .fraction_field import FractionField
+from sage.rings.fraction_field import FractionField
 Frac = FractionField
 
 # Localization
-from .localization import Localization
+from sage.rings.localization import Localization
 
 # c-finite sequences
-from .cfinite_sequence import CFiniteSequence, CFiniteSequences
+from sage.rings.cfinite_sequence import CFiniteSequence, CFiniteSequences
 
-from .bernoulli_mod_p import bernoulli_mod_p, bernoulli_mod_p_single
+from sage.rings.bernoulli_mod_p import bernoulli_mod_p, bernoulli_mod_p_single
 
-from .monomials import monomials
+from sage.rings.monomials import monomials
 
-from .cc import CC
-from .cif import CIF
+from sage.rings.cc import CC
+from sage.rings.cif import CIF
 
 # invariant theory
-from .invariants.all import *
+from sage.rings.invariants.all import *
 
-from .fast_arith import prime_range
+from sage.rings.fast_arith import prime_range
 
 # continued fractions
 from sage.rings.continued_fraction import (continued_fraction,
                                            continued_fraction_list)
 
 # asymptotic ring
-from .asymptotic.all import *
+from sage.rings.asymptotic.all import *
 
 # Register classes in numbers abc
-from . import numbers_abc
+from sage.rings import numbers_abc
diff --git a/src/sage/rings/all__sagemath_categories.py b/src/sage/rings/all__sagemath_categories.py
index 8d14e7afcda..d03d1c6b483 100644
--- a/src/sage/rings/all__sagemath_categories.py
+++ b/src/sage/rings/all__sagemath_categories.py
@@ -1,6 +1,6 @@
 # Ring base classes
-from .ring import Ring
-
+from sage.rings.ring import Ring
 # Ideals
-from .ideal import Ideal
+from sage.rings.ideal import Ideal
+
 ideal = Ideal
diff --git a/src/sage/rings/big_oh.py b/src/sage/rings/big_oh.py
index e9b22e77b37..917c81e0a3d 100644
--- a/src/sage/rings/big_oh.py
+++ b/src/sage/rings/big_oh.py
@@ -11,6 +11,7 @@
 
 from sage.arith.misc import factor
 from sage.misc.lazy_import import lazy_import
+
 lazy_import('sage.rings.padics.factory', ['Qp', 'Zp'])
 lazy_import('sage.rings.padics.padic_generic_element', 'pAdicGenericElement')
 from sage.rings.polynomial.polynomial_element import Polynomial
@@ -25,10 +26,12 @@
 except ImportError:
     PuiseuxSeries = ()
 
-from . import power_series_ring_element
-from . import integer
-from . import rational
-from . import multi_power_series_ring_element
+from sage.rings import (
+    integer,
+    multi_power_series_ring_element,
+    power_series_ring_element,
+    rational,
+)
 
 
 def O(*x, **kwds):
diff --git a/src/sage/rings/cc.py b/src/sage/rings/cc.py
index 6db89579029..94e3093488b 100644
--- a/src/sage/rings/cc.py
+++ b/src/sage/rings/cc.py
@@ -1,3 +1,3 @@
-from .complex_mpfr import ComplexField
+from sage.rings.complex_mpfr import ComplexField
 
 CC = ComplexField()
diff --git a/src/sage/rings/cif.py b/src/sage/rings/cif.py
index 91924f4d26c..900c635e87b 100644
--- a/src/sage/rings/cif.py
+++ b/src/sage/rings/cif.py
@@ -1,3 +1,3 @@
-from .complex_interval_field import ComplexIntervalField
+from sage.rings.complex_interval_field import ComplexIntervalField
 
 CIF = ComplexIntervalField()
diff --git a/src/sage/rings/complex_conversion.pyx b/src/sage/rings/complex_conversion.pyx
index e7e53724f9c..39c8b63eb55 100644
--- a/src/sage/rings/complex_conversion.pyx
+++ b/src/sage/rings/complex_conversion.pyx
@@ -1,5 +1,5 @@
-from .complex_double cimport ComplexDoubleElement
-from .complex_mpfr cimport ComplexNumber
+from sage.rings.complex_double cimport ComplexDoubleElement
+from sage.rings.complex_mpfr cimport ComplexNumber
 from sage.libs.mpfr cimport mpfr_get_d, MPFR_RNDN
 from sage.libs.gsl.complex cimport GSL_SET_COMPLEX
 
diff --git a/src/sage/rings/complex_double.pyx b/src/sage/rings/complex_double.pyx
index 8f1e8205817..944bcc5aa27 100644
--- a/src/sage/rings/complex_double.pyx
+++ b/src/sage/rings/complex_double.pyx
@@ -105,10 +105,10 @@ complex_double_element_gamma = None
 complex_double_element_gamma_inc = None
 complex_double_element_zeta = None
 
-from .complex_conversion cimport CCtoCDF
+from sage.rings.complex_conversion cimport CCtoCDF
 
-from .real_double cimport RealDoubleElement, double_repr
-from .real_double import RDF
+from sage.rings.real_double cimport RealDoubleElement, double_repr
+from sage.rings.real_double import RDF
 from sage.rings.integer_ring import ZZ
 from sage.structure.richcmp cimport rich_to_bool
 
diff --git a/src/sage/rings/complex_interval.pxd b/src/sage/rings/complex_interval.pxd
index 56513b65747..60d8c010cd8 100644
--- a/src/sage/rings/complex_interval.pxd
+++ b/src/sage/rings/complex_interval.pxd
@@ -2,7 +2,7 @@ from sage.libs.mpfr.types cimport mpfr_prec_t
 from sage.libs.mpfi.types cimport mpfi_t
 
 cimport sage.structure.element
-from .real_mpfi cimport RealIntervalFieldElement, RealIntervalField_class
+from sage.rings.real_mpfi cimport RealIntervalFieldElement, RealIntervalField_class
 
 
 cdef class ComplexIntervalFieldElement(sage.structure.element.FieldElement):
diff --git a/src/sage/rings/complex_interval.pyx b/src/sage/rings/complex_interval.pyx
index c38261d136d..5b7a04fef10 100644
--- a/src/sage/rings/complex_interval.pyx
+++ b/src/sage/rings/complex_interval.pyx
@@ -66,12 +66,12 @@ from sage.arith.constants cimport LOG_TEN_TWO_PLUS_EPSILON
 
 from sage.structure.element cimport FieldElement
 from sage.structure.parent cimport Parent
-from .complex_mpfr cimport ComplexNumber
+from sage.rings.complex_mpfr cimport ComplexNumber
 from sage.rings.integer cimport Integer
 cimport sage.rings.real_mpfi as real_mpfi
-from .real_mpfr cimport RealNumber
-from .convert.mpfi cimport mpfi_set_sage
-from .infinity import infinity
+from sage.rings.real_mpfr cimport RealNumber
+from sage.rings.convert.mpfi cimport mpfi_set_sage
+from sage.rings.infinity import infinity
 
 
 def is_ComplexIntervalFieldElement(x):
diff --git a/src/sage/rings/complex_interval_field.py b/src/sage/rings/complex_interval_field.py
index 7903ab42bae..c2524923459 100644
--- a/src/sage/rings/complex_interval_field.py
+++ b/src/sage/rings/complex_interval_field.py
@@ -36,18 +36,17 @@
 # ****************************************************************************
 
 
-from sage.structure.parent import Parent
-from .integer_ring import ZZ
-from .rational_field import QQ
-from .ring import Field
-import sage.rings.abc
-from . import integer
-from . import complex_interval
 import weakref
-from .real_mpfi import RealIntervalField, RealIntervalField_class
-from .complex_mpfr import ComplexField
-from sage.misc.cachefunc import cached_method
 
+import sage.rings.abc
+from sage.misc.cachefunc import cached_method
+from sage.rings import complex_interval, integer
+from sage.rings.complex_mpfr import ComplexField
+from sage.rings.integer_ring import ZZ
+from sage.rings.rational_field import QQ
+from sage.rings.real_mpfi import RealIntervalField, RealIntervalField_class
+from sage.rings.ring import Field
+from sage.structure.parent import Parent
 
 cache = {}
 def ComplexIntervalField(prec=53, names=None):
diff --git a/src/sage/rings/complex_mpc.pyx b/src/sage/rings/complex_mpc.pyx
index 562286db44e..7a9ab37e268 100644
--- a/src/sage/rings/complex_mpc.pyx
+++ b/src/sage/rings/complex_mpc.pyx
@@ -59,7 +59,7 @@ EXAMPLES::
 # ****************************************************************************
 
 import re
-from . import real_mpfr
+from sage.rings import real_mpfr
 import weakref
 from cpython.object cimport Py_NE
 
@@ -74,13 +74,13 @@ from sage.structure.richcmp cimport rich_to_bool
 from sage.categories.map cimport Map
 from sage.libs.pari.all import pari
 
-from .integer cimport Integer
-from .complex_mpfr cimport ComplexNumber
-from .complex_mpfr import ComplexField_class
+from sage.rings.integer cimport Integer
+from sage.rings.complex_mpfr cimport ComplexNumber
+from sage.rings.complex_mpfr import ComplexField_class
 
 from sage.misc.randstate cimport randstate, current_randstate
-from .real_mpfr cimport RealField_class, RealNumber
-from .real_mpfr import mpfr_prec_min, mpfr_prec_max
+from sage.rings.real_mpfr cimport RealField_class, RealNumber
+from sage.rings.real_mpfr import mpfr_prec_min, mpfr_prec_max
 from sage.structure.richcmp cimport rich_to_bool, richcmp
 from sage.categories.fields import Fields
 
diff --git a/src/sage/rings/complex_mpfr.pxd b/src/sage/rings/complex_mpfr.pxd
index 4aa6de62a69..95972c52bb5 100644
--- a/src/sage/rings/complex_mpfr.pxd
+++ b/src/sage/rings/complex_mpfr.pxd
@@ -1,7 +1,7 @@
 from sage.libs.mpfr.types cimport mpfr_t, mpfr_prec_t
 
 cimport sage.structure.element
-from .real_mpfr cimport RealNumber
+from sage.rings.real_mpfr cimport RealNumber
 
 cdef class ComplexNumber(sage.structure.element.FieldElement):
     cdef mpfr_t __re
diff --git a/src/sage/rings/complex_mpfr.pyx b/src/sage/rings/complex_mpfr.pyx
index d6f92bb2fee..c8a27ac7aec 100644
--- a/src/sage/rings/complex_mpfr.pyx
+++ b/src/sage/rings/complex_mpfr.pyx
@@ -46,11 +46,11 @@ from sage.misc.sage_eval import sage_eval
 
 import sage.rings.abc
 from sage.arith.constants cimport LOG_TEN_TWO_PLUS_EPSILON
-from . import infinity
-from .integer cimport Integer
+from sage.rings import infinity
+from sage.rings.integer cimport Integer
 
-from .complex_double cimport ComplexDoubleElement
-from .real_mpfr cimport RealNumber
+from sage.rings.complex_double cimport ComplexDoubleElement
+from sage.rings.real_mpfr cimport RealNumber
 from sage.libs.gsl.complex cimport *
 
 from sage.libs.mpmath.utils cimport mpfr_to_mpfval
diff --git a/src/sage/rings/continued_fraction.py b/src/sage/rings/continued_fraction.py
index 1980d55b27c..a813816ac50 100644
--- a/src/sage/rings/continued_fraction.py
+++ b/src/sage/rings/continued_fraction.py
@@ -207,12 +207,12 @@
 
 import numbers
 
-from sage.structure.sage_object import SageObject
-from sage.structure.richcmp import richcmp_method, rich_to_bool
 import sage.rings.abc
-from .integer import Integer
-from .integer_ring import ZZ
-from .infinity import Infinity
+from sage.rings.infinity import Infinity
+from sage.rings.integer import Integer
+from sage.rings.integer_ring import ZZ
+from sage.structure.richcmp import rich_to_bool, richcmp_method
+from sage.structure.sage_object import SageObject
 
 ZZ_0 = Integer(0)
 ZZ_1 = Integer(1)
@@ -849,8 +849,9 @@ def convergents(self):
             Add an example with infinite list.
         """
         if self.length() == Infinity:
-            from sage.misc.lazy_list import lazy_list
             from itertools import count
+
+            from sage.misc.lazy_list import lazy_list
             return lazy_list(self.numerator(n) / self.denominator(n)
                              for n in count())
         return [self.numerator(n) / self.denominator(n)
@@ -876,8 +877,9 @@ def quotients(self):
             Add an example with infinite list.
         """
         if self.length() == Infinity:
-            from sage.misc.lazy_list import lazy_list
             from itertools import count
+
+            from sage.misc.lazy_list import lazy_list
             return lazy_list(self.quotient(n) for n in count())
         return [self.quotient(n) for n in range(len(self))]
 
@@ -1147,8 +1149,7 @@ def numerical_approx(self, prec=None, digits=None, algorithm=None):
             sage: cf.n(digits=33)                                                       # needs sage.combinat
             1.28102513329556981555293038097590
         """
-        from sage.arith.numerical_approx import (digits_to_bits,
-                                                 numerical_approx_generic)
+        from sage.arith.numerical_approx import digits_to_bits, numerical_approx_generic
         if prec is None:
             prec = digits_to_bits(digits)
         return numerical_approx_generic(self, prec)
@@ -1564,8 +1565,8 @@ def value(self):
 
         # now x is one of the root of the equation
         #   q1 x^2 + (q0 - p1) x - p0 = 0
-        from sage.rings.number_field.number_field import QuadraticField
         from sage.misc.functional import squarefree_part
+        from sage.rings.number_field.number_field import QuadraticField
         D = (q0-p1)**2 + 4*q1*p0
         DD = squarefree_part(D)
         Q = QuadraticField(DD, 'sqrt%d' % DD)
@@ -2674,8 +2675,8 @@ def continued_fraction(x, value=None):
         return ContinuedFraction_periodic(x1, x2)
 
     # input for infinite partial quotient expansion
-    from sage.misc.lazy_list import lazy_list_generic
     from sage.combinat.words.infinite_word import InfiniteWord_class
+    from sage.misc.lazy_list import lazy_list_generic
     if isinstance(x, (lazy_list_generic, InfiniteWord_class)):
         return ContinuedFraction_infinite(x, value)
 
diff --git a/src/sage/rings/convert/mpfi.pyx b/src/sage/rings/convert/mpfi.pyx
index abef4963bcf..0d42927740a 100644
--- a/src/sage/rings/convert/mpfi.pyx
+++ b/src/sage/rings/convert/mpfi.pyx
@@ -23,14 +23,14 @@ from sage.cpython.string cimport bytes_to_str
 from sage.structure.element cimport Element
 
 import sage.rings.abc
-from ..integer cimport Integer
-from ..rational cimport Rational
-from ..real_mpfi cimport RealIntervalFieldElement, RealIntervalField_class
-from ..real_mpfr cimport RealNumber
-from ..real_double cimport RealDoubleElement
-from ..complex_mpfr cimport ComplexNumber
-from ..complex_interval cimport ComplexIntervalFieldElement
-from ..complex_double cimport ComplexDoubleElement
+from sage.rings.integer cimport Integer
+from sage.rings.rational cimport Rational
+from sage.rings.real_mpfi cimport RealIntervalFieldElement, RealIntervalField_class
+from sage.rings.real_mpfr cimport RealNumber
+from sage.rings.real_double cimport RealDoubleElement
+from sage.rings.complex_mpfr cimport ComplexNumber
+from sage.rings.complex_interval cimport ComplexIntervalFieldElement
+from sage.rings.complex_double cimport ComplexDoubleElement
 
 from cypari2.gen cimport Gen
 
diff --git a/src/sage/rings/finite_rings/element_givaro.pyx b/src/sage/rings/finite_rings/element_givaro.pyx
index 0483b4949e7..7619ec7185f 100644
--- a/src/sage/rings/finite_rings/element_givaro.pyx
+++ b/src/sage/rings/finite_rings/element_givaro.pyx
@@ -57,7 +57,7 @@ from cysignals.signals cimport sig_on, sig_off
 from cypari2.paridecl cimport *
 
 from sage.misc.randstate cimport current_randstate
-from .element_pari_ffelt cimport FiniteFieldElement_pari_ffelt
+from sage.rings.finite_rings.element_pari_ffelt cimport FiniteFieldElement_pari_ffelt
 from sage.structure.richcmp cimport richcmp
 import sage.arith.all
 
diff --git a/src/sage/rings/finite_rings/element_ntl_gf2e.pyx b/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
index 61e7b05048c..60d275ea5ae 100644
--- a/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
+++ b/src/sage/rings/finite_rings/element_ntl_gf2e.pyx
@@ -44,8 +44,8 @@ from sage.libs.pari.all import pari
 from cypari2.gen cimport Gen
 from cypari2.stack cimport clear_stack
 
-from .element_pari_ffelt import FiniteFieldElement_pari_ffelt
-from .finite_field_ntl_gf2e import FiniteField_ntl_gf2e
+from sage.rings.finite_rings.element_pari_ffelt import FiniteFieldElement_pari_ffelt
+from sage.rings.finite_rings.finite_field_ntl_gf2e import FiniteField_ntl_gf2e
 
 from sage.interfaces.abc import GapElement
 
diff --git a/src/sage/rings/finite_rings/element_pari_ffelt.pyx b/src/sage/rings/finite_rings/element_pari_ffelt.pyx
index 9beb505cae3..ca262f6f668 100644
--- a/src/sage/rings/finite_rings/element_pari_ffelt.pyx
+++ b/src/sage/rings/finite_rings/element_pari_ffelt.pyx
@@ -24,8 +24,8 @@ from sage.libs.pari.convert_gmp cimport _new_GEN_from_mpz_t
 from cypari2.stack cimport new_gen, new_gen_noclear, clear_stack
 from cypari2.gen cimport Gen as pari_gen, objtogen
 
-from .element_base cimport FinitePolyExtElement
-from .integer_mod import IntegerMod_abstract
+from sage.rings.finite_rings.element_base cimport FinitePolyExtElement
+from sage.rings.finite_rings.integer_mod import IntegerMod_abstract
 
 import sage.rings.integer
 from sage.rings.integer cimport Integer
diff --git a/src/sage/rings/finite_rings/hom_finite_field_givaro.pxd b/src/sage/rings/finite_rings/hom_finite_field_givaro.pxd
index 62898386212..f5a70c975f9 100644
--- a/src/sage/rings/finite_rings/hom_finite_field_givaro.pxd
+++ b/src/sage/rings/finite_rings/hom_finite_field_givaro.pxd
@@ -1,8 +1,8 @@
-from .hom_finite_field cimport (SectionFiniteFieldHomomorphism_generic,
+from sage.rings.finite_rings.hom_finite_field cimport (SectionFiniteFieldHomomorphism_generic,
     FiniteFieldHomomorphism_generic, FrobeniusEndomorphism_finite_field)
 
 from sage.structure.element cimport Element
-from .element_givaro cimport Cache_givaro
+from sage.rings.finite_rings.element_givaro cimport Cache_givaro
 
 
 cdef class SectionFiniteFieldHomomorphism_givaro(SectionFiniteFieldHomomorphism_generic):
diff --git a/src/sage/rings/finite_rings/hom_finite_field_givaro.pyx b/src/sage/rings/finite_rings/hom_finite_field_givaro.pyx
index 21036266df1..bc685731610 100644
--- a/src/sage/rings/finite_rings/hom_finite_field_givaro.pyx
+++ b/src/sage/rings/finite_rings/hom_finite_field_givaro.pyx
@@ -33,21 +33,21 @@ AUTHOR:
 
 from sage.rings.finite_rings.finite_field_constructor import FiniteField
 
-from .hom_finite_field cimport SectionFiniteFieldHomomorphism_generic
-from .hom_finite_field cimport FiniteFieldHomomorphism_generic
-from .hom_finite_field cimport FrobeniusEndomorphism_finite_field
+from sage.rings.finite_rings.hom_finite_field cimport SectionFiniteFieldHomomorphism_generic
+from sage.rings.finite_rings.hom_finite_field cimport FiniteFieldHomomorphism_generic
+from sage.rings.finite_rings.hom_finite_field cimport FrobeniusEndomorphism_finite_field
 
-from .hom_prime_finite_field cimport FiniteFieldHomomorphism_prime
+from sage.rings.finite_rings.hom_prime_finite_field cimport FiniteFieldHomomorphism_prime
 
 from sage.categories.homset import Hom
 from sage.structure.element cimport Element
 
 from sage.rings.finite_rings.finite_field_givaro import FiniteField_givaro
-from .element_givaro cimport FiniteField_givaroElement
+from sage.rings.finite_rings.element_givaro cimport FiniteField_givaroElement
 #from element_givaro cimport make_FiniteField_givaroElement
 
 from sage.structure.parent cimport Parent
-from .element_givaro cimport Cache_givaro
+from sage.rings.finite_rings.element_givaro cimport Cache_givaro
 
 
 cdef class SectionFiniteFieldHomomorphism_givaro(SectionFiniteFieldHomomorphism_generic):
diff --git a/src/sage/rings/finite_rings/hom_prime_finite_field.pxd b/src/sage/rings/finite_rings/hom_prime_finite_field.pxd
index c0da361fb42..b146b0c9169 100644
--- a/src/sage/rings/finite_rings/hom_prime_finite_field.pxd
+++ b/src/sage/rings/finite_rings/hom_prime_finite_field.pxd
@@ -1,4 +1,4 @@
-from .hom_finite_field cimport (SectionFiniteFieldHomomorphism_generic,
+from sage.rings.finite_rings.hom_finite_field cimport (SectionFiniteFieldHomomorphism_generic,
     FiniteFieldHomomorphism_generic, FrobeniusEndomorphism_finite_field)
 
 
diff --git a/src/sage/rings/finite_rings/hom_prime_finite_field.pyx b/src/sage/rings/finite_rings/hom_prime_finite_field.pyx
index 6bc38546612..2ff550aa1b1 100644
--- a/src/sage/rings/finite_rings/hom_prime_finite_field.pyx
+++ b/src/sage/rings/finite_rings/hom_prime_finite_field.pyx
@@ -27,9 +27,9 @@ AUTHOR:
 from sage.categories.homset import Hom
 from sage.structure.element cimport Element
 
-from .hom_finite_field cimport SectionFiniteFieldHomomorphism_generic
-from .hom_finite_field cimport FiniteFieldHomomorphism_generic
-from .hom_finite_field cimport FrobeniusEndomorphism_finite_field
+from sage.rings.finite_rings.hom_finite_field cimport SectionFiniteFieldHomomorphism_generic
+from sage.rings.finite_rings.hom_finite_field cimport FiniteFieldHomomorphism_generic
+from sage.rings.finite_rings.hom_finite_field cimport FrobeniusEndomorphism_finite_field
 
 from sage.rings.finite_rings.finite_field_base import FiniteField
 
diff --git a/src/sage/rings/finite_rings/integer_mod_ring.py b/src/sage/rings/finite_rings/integer_mod_ring.py
index a448046f137..0bb4ce21f4b 100644
--- a/src/sage/rings/finite_rings/integer_mod_ring.py
+++ b/src/sage/rings/finite_rings/integer_mod_ring.py
@@ -67,7 +67,7 @@
 from sage.arith.misc import CRT_basis
 import sage.rings.ring as ring
 import sage.rings.abc
-from . import integer_mod
+from sage.rings.finite_rings import integer_mod
 import sage.rings.integer as integer
 import sage.rings.integer_ring as integer_ring
 import sage.rings.quotient_ring as quotient_ring
diff --git a/src/sage/rings/fraction_field.py b/src/sage/rings/fraction_field.py
index 5d298720a90..bfa690cec0e 100644
--- a/src/sage/rings/fraction_field.py
+++ b/src/sage/rings/fraction_field.py
@@ -83,19 +83,17 @@
 #                  http://www.gnu.org/licenses/
 # ****************************************************************************
 
-from . import ring
-from . import fraction_field_element
 import sage.misc.latex as latex
+from sage.categories.basic import QuotientFields, Rings
+from sage.categories.map import Section
 from sage.misc.cachefunc import cached_method
-
+from sage.rings import fraction_field_element, ring
 from sage.rings.integer_ring import ZZ
-from sage.structure.richcmp import richcmp
-from sage.structure.parent import Parent
-from sage.structure.element import parent
 from sage.structure.coerce import py_scalar_to_element
 from sage.structure.coerce_maps import CallableConvertMap, DefaultConvertMap_unique
-from sage.categories.basic import QuotientFields, Rings
-from sage.categories.map import Section
+from sage.structure.element import parent
+from sage.structure.parent import Parent
+from sage.structure.richcmp import richcmp
 
 
 def FractionField(R, names=None):
@@ -327,10 +325,11 @@ def _coerce_map_from_(self, S):
             sage: f(L(1/7)) == 1/7
             True
         """
-        from sage.rings.rational_field import QQ
         from sage.rings.number_field.number_field_base import NumberField
-        from sage.rings.polynomial.laurent_polynomial_ring_base import \
-            LaurentPolynomialRing_generic
+        from sage.rings.polynomial.laurent_polynomial_ring_base import (
+            LaurentPolynomialRing_generic,
+        )
+        from sage.rings.rational_field import QQ
 
         if S is self._R:
             parent = self._R.Hom(self)
@@ -1079,7 +1078,9 @@ def _coerce_map_from_(self, R):
             1/t
 
         """
-        from sage.rings.function_field.function_field_rational import RationalFunctionField
+        from sage.rings.function_field.function_field_rational import (
+            RationalFunctionField,
+        )
         if isinstance(R, RationalFunctionField) and self.variable_name() == R.variable_name() and self.base_ring() is R.constant_base_field():
             from sage.categories.homset import Hom
             parent = Hom(R, self)
@@ -1160,8 +1161,8 @@ def section(self):
               To:   Univariate Polynomial Ring in x over Rational Field
 
         """
-        from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
         from sage.categories.homset import Hom
+        from sage.categories.sets_with_partial_maps import SetsWithPartialMaps
         parent = Hom(self.codomain(), self.domain(), SetsWithPartialMaps())
         return parent.__make_element_class__(FractionFieldEmbeddingSection)(self)
 
diff --git a/src/sage/rings/fraction_field_element.pyx b/src/sage/rings/fraction_field_element.pyx
index cb0bbce912f..5b6cfa2e29f 100644
--- a/src/sage/rings/fraction_field_element.pyx
+++ b/src/sage/rings/fraction_field_element.pyx
@@ -22,7 +22,7 @@ AUTHORS:
 from sage.structure.element cimport FieldElement, parent
 from sage.structure.richcmp cimport richcmp
 
-from .rational_field import QQ
+from sage.rings.rational_field import QQ
 
 import sage.misc.latex as latex
 
diff --git a/src/sage/rings/homset.py b/src/sage/rings/homset.py
index 0e500de2608..b2f24a3800a 100644
--- a/src/sage/rings/homset.py
+++ b/src/sage/rings/homset.py
@@ -12,10 +12,11 @@
 
 from sage.categories.homset import HomsetWithBase
 from sage.categories.rings import Rings
+
 _Rings = Rings()
 
-from . import morphism
-from . import quotient_ring
+from sage.rings import morphism, quotient_ring
+
 
 def is_RingHomset(H):
     """
diff --git a/src/sage/rings/ideal_monoid.py b/src/sage/rings/ideal_monoid.py
index 0988ffc72e5..019348a6afe 100644
--- a/src/sage/rings/ideal_monoid.py
+++ b/src/sage/rings/ideal_monoid.py
@@ -8,10 +8,10 @@
     <class 'sage.rings.ideal_monoid.IdealMonoid_c_with_category'>
 """
 
-from sage.structure.parent import Parent
 import sage.rings.integer_ring
-from . import ideal
 from sage.categories.monoids import Monoids
+from sage.rings import ideal
+from sage.structure.parent import Parent
 
 
 def IdealMonoid(R):
diff --git a/src/sage/rings/imaginary_unit.py b/src/sage/rings/imaginary_unit.py
index 09349e66e8b..bb29cd32630 100644
--- a/src/sage/rings/imaginary_unit.py
+++ b/src/sage/rings/imaginary_unit.py
@@ -1,5 +1,5 @@
 # coding: utf-8
 
-from .number_field.number_field import GaussianField
+from sage.rings.number_field.number_field import GaussianField
 
 I = GaussianField().gen()
diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index 3ecd0bd986d..143641e4472 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -183,7 +183,7 @@ from sage.structure.element import coerce_binop
 
 from sage.structure.richcmp cimport rich_to_bool_sgn
 
-from . import integer_ring
+from sage.rings import integer_ring
 
 cimport gmpy2
 gmpy2.import_gmpy2()
diff --git a/src/sage/rings/integer_ring.pxd b/src/sage/rings/integer_ring.pxd
index 41afccaa514..d0af1bc068f 100644
--- a/src/sage/rings/integer_ring.pxd
+++ b/src/sage/rings/integer_ring.pxd
@@ -1,5 +1,5 @@
-from .ring cimport PrincipalIdealDomain
-from .integer cimport Integer
+from sage.rings.ring cimport PrincipalIdealDomain
+from sage.rings.integer cimport Integer
 from sage.libs.gmp.types cimport mpz_t
 
 cdef class IntegerRing_class(PrincipalIdealDomain):
diff --git a/src/sage/rings/integer_ring.pyx b/src/sage/rings/integer_ring.pyx
index b510c4ba992..1ee45b99a59 100644
--- a/src/sage/rings/integer_ring.pyx
+++ b/src/sage/rings/integer_ring.pyx
@@ -69,7 +69,7 @@ from sage.misc.randstate cimport randstate, current_randstate, SAGE_RAND_MAX
 
 cimport sage.rings.integer as integer
 
-from . import ring
+from sage.rings import ring
 
 arith = None
 cdef void late_import() noexcept:
diff --git a/src/sage/rings/laurent_series_ring.py b/src/sage/rings/laurent_series_ring.py
index 565e5d12f4c..187ac847522 100644
--- a/src/sage/rings/laurent_series_ring.py
+++ b/src/sage/rings/laurent_series_ring.py
@@ -32,20 +32,17 @@
 # ****************************************************************************
 
 
-from sage.categories.rings import Rings
-from sage.rings.infinity import infinity
 from sage.categories.algebras import Algebras
-from sage.categories.integral_domains import IntegralDomains
-from sage.categories.fields import Fields
 from sage.categories.complete_discrete_valuation import CompleteDiscreteValuationFields
-
-from .laurent_series_ring_element import LaurentSeries
-from .ring import CommutativeRing
-
-from sage.structure.unique_representation import UniqueRepresentation
+from sage.categories.fields import Fields
+from sage.categories.integral_domains import IntegralDomains
+from sage.categories.rings import Rings
 from sage.misc.cachefunc import cached_method
-
+from sage.rings.infinity import infinity
 from sage.rings.integer_ring import ZZ
+from sage.rings.laurent_series_ring_element import LaurentSeries
+from sage.rings.ring import CommutativeRing
+from sage.structure.unique_representation import UniqueRepresentation
 
 try:
     from sage.libs.pari.all import pari_gen
@@ -315,8 +312,8 @@ def fraction_field(self):
             ...
             ValueError: must be an integral domain
         """
-        from sage.categories.integral_domains import IntegralDomains
         from sage.categories.fields import Fields
+        from sage.categories.integral_domains import IntegralDomains
         if self in Fields():
             return self
         elif self in IntegralDomains():
@@ -489,8 +486,8 @@ def _element_constructor_(self, x, n=0, prec=infinity):
             x^-3
         """
         from sage.rings.fraction_field_element import is_FractionFieldElement
-        from sage.rings.polynomial.polynomial_element import Polynomial
         from sage.rings.polynomial.multi_polynomial import MPolynomial
+        from sage.rings.polynomial.polynomial_element import Polynomial
         from sage.structure.element import parent
 
         P = parent(x)
@@ -643,9 +640,11 @@ def _coerce_map_from_(self, P):
             True
         """
         A = self.base_ring()
+        from sage.rings.polynomial.laurent_polynomial_ring_base import (
+            LaurentPolynomialRing_generic,
+        )
         from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
         from sage.rings.power_series_ring import is_PowerSeriesRing
-        from sage.rings.polynomial.laurent_polynomial_ring_base import LaurentPolynomialRing_generic
 
         if ((is_LaurentSeriesRing(P) or
              isinstance(P, LaurentPolynomialRing_generic) or
diff --git a/src/sage/rings/laurent_series_ring_element.pyx b/src/sage/rings/laurent_series_ring_element.pyx
index b15e6cbdcf0..cac7acd812c 100644
--- a/src/sage/rings/laurent_series_ring_element.pyx
+++ b/src/sage/rings/laurent_series_ring_element.pyx
@@ -66,12 +66,12 @@ AUTHORS:
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from .infinity import infinity
+from sage.rings.infinity import infinity
 
 from sage.rings.rational_field import QQ
 import sage.misc.latex
 from sage.rings.polynomial.laurent_polynomial import LaurentPolynomial_univariate
-from .power_series_ring_element cimport PowerSeries
+from sage.rings.power_series_ring_element cimport PowerSeries
 from sage.structure.element cimport Element, AlgebraElement
 from sage.structure.richcmp cimport richcmp_not_equal, rich_to_bool
 from sage.misc.derivative import multi_derivative
diff --git a/src/sage/rings/morphism.pyx b/src/sage/rings/morphism.pyx
index 9901eff95a3..f8e32bd8c62 100644
--- a/src/sage/rings/morphism.pyx
+++ b/src/sage/rings/morphism.pyx
@@ -406,7 +406,7 @@ compare equal::
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
-from . import ideal
+from sage.rings import ideal
 import sage.structure.all
 from sage.structure.richcmp cimport (richcmp, rich_to_bool)
 from sage.misc.cachefunc import cached_method
diff --git a/src/sage/rings/number_field/number_field_element_quadratic.pxd b/src/sage/rings/number_field/number_field_element_quadratic.pxd
index 76661971848..b9fbb294a54 100644
--- a/src/sage/rings/number_field/number_field_element_quadratic.pxd
+++ b/src/sage/rings/number_field/number_field_element_quadratic.pxd
@@ -2,7 +2,7 @@ from sage.libs.gmp.types cimport mpz_t
 from sage.libs.arb.types cimport arb_t
 from sage.rings.integer cimport Integer
 from sage.rings.rational cimport Rational
-from .number_field_element cimport NumberFieldElement, NumberFieldElement_absolute
+from sage.rings.number_field.number_field_element cimport NumberFieldElement, NumberFieldElement_absolute
 
 
 cdef class NumberFieldElement_quadratic(NumberFieldElement_absolute):
diff --git a/src/sage/rings/polynomial/multi_polynomial.pxd b/src/sage/rings/polynomial/multi_polynomial.pxd
index 73bde26ab51..3f4fe16888c 100644
--- a/src/sage/rings/polynomial/multi_polynomial.pxd
+++ b/src/sage/rings/polynomial/multi_polynomial.pxd
@@ -1,4 +1,4 @@
-from .commutative_polynomial cimport CommutativePolynomial
+from sage.rings.polynomial.commutative_polynomial cimport CommutativePolynomial
 
 
 cdef class MPolynomial(CommutativePolynomial):
diff --git a/src/sage/rings/polynomial/polynomial_element.pxd b/src/sage/rings/polynomial/polynomial_element.pxd
index b337919a807..9b8bf41fa06 100644
--- a/src/sage/rings/polynomial/polynomial_element.pxd
+++ b/src/sage/rings/polynomial/polynomial_element.pxd
@@ -2,8 +2,8 @@ from sage.structure.element import Element
 from sage.structure.element cimport Element, CommutativeAlgebraElement, ModuleElement
 from sage.structure.parent cimport Parent
 from sage.rings.integer cimport Integer
-from .commutative_polynomial cimport CommutativePolynomial
-from .polynomial_compiled cimport CompiledPolynomialFunction
+from sage.rings.polynomial.commutative_polynomial cimport CommutativePolynomial
+from sage.rings.polynomial.polynomial_compiled cimport CompiledPolynomialFunction
 
 
 cdef class Polynomial(CommutativePolynomial):
diff --git a/src/sage/rings/polynomial/polynomial_element.pyx b/src/sage/rings/polynomial/polynomial_element.pyx
index 8e9c4822a7b..ede464edddf 100644
--- a/src/sage/rings/polynomial/polynomial_element.pyx
+++ b/src/sage/rings/polynomial/polynomial_element.pyx
@@ -118,7 +118,7 @@ from sage.misc.derivative import multi_derivative
 from sage.arith.misc import sort_complex_numbers_for_display, power_mod, is_prime
 from sage.arith.functions import lcm
 
-from . import polynomial_fateman
+from sage.rings.polynomial import polynomial_fateman
 
 from sage.rings.ideal import is_Ideal
 from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
@@ -182,7 +182,7 @@ cpdef is_Polynomial(f) noexcept:
 
     return isinstance(f, Polynomial)
 
-from .polynomial_compiled cimport CompiledPolynomialFunction
+from sage.rings.polynomial.polynomial_compiled cimport CompiledPolynomialFunction
 
 from sage.rings.polynomial.polydict cimport ETuple
 
diff --git a/src/sage/rings/polynomial/polynomial_real_mpfr_dense.pyx b/src/sage/rings/polynomial/polynomial_real_mpfr_dense.pyx
index 15c65081cd9..8bd5bd00a67 100644
--- a/src/sage/rings/polynomial/polynomial_real_mpfr_dense.pyx
+++ b/src/sage/rings/polynomial/polynomial_real_mpfr_dense.pyx
@@ -30,7 +30,7 @@ from cpython.long cimport PyLong_AsLong
 from cpython.float cimport PyFloat_AS_DOUBLE
 
 from sage.structure.parent cimport Parent
-from .polynomial_element cimport Polynomial, _dict_to_list
+from sage.rings.polynomial.polynomial_element cimport Polynomial, _dict_to_list
 from sage.rings.real_mpfr cimport RealField_class, RealNumber
 from sage.rings.integer cimport Integer, smallInteger
 from sage.rings.rational cimport Rational
diff --git a/src/sage/rings/power_series_mpoly.pxd b/src/sage/rings/power_series_mpoly.pxd
index d358ada1356..d8816e694b7 100644
--- a/src/sage/rings/power_series_mpoly.pxd
+++ b/src/sage/rings/power_series_mpoly.pxd
@@ -1,5 +1,5 @@
 from sage.structure.element cimport ModuleElement
-from .power_series_ring_element cimport PowerSeries
+from sage.rings.power_series_ring_element cimport PowerSeries
 
 cdef class PowerSeries_mpoly(PowerSeries):
     cdef ModuleElement __f
diff --git a/src/sage/rings/power_series_mpoly.pyx b/src/sage/rings/power_series_mpoly.pyx
index fa62a7ef6d6..d11eec97269 100644
--- a/src/sage/rings/power_series_mpoly.pyx
+++ b/src/sage/rings/power_series_mpoly.pyx
@@ -1,10 +1,10 @@
 # NOT ready to be used -- possibly should be deleted.
 
-from .power_series_ring_element cimport PowerSeries
+from sage.rings.power_series_ring_element cimport PowerSeries
 from sage.structure.element cimport Element
-from .infinity import infinity
-from .polynomial.multi_polynomial_ring_base import is_MPolynomialRing
-from . import power_series_poly
+from sage.rings.infinity import infinity
+from sage.rings.polynomial.multi_polynomial_ring_base import is_MPolynomialRing
+from sage.rings import power_series_poly
 
 
 cdef class PowerSeries_mpoly(PowerSeries):
diff --git a/src/sage/rings/power_series_pari.pxd b/src/sage/rings/power_series_pari.pxd
index fadf7f0fd08..35b37929c33 100644
--- a/src/sage/rings/power_series_pari.pxd
+++ b/src/sage/rings/power_series_pari.pxd
@@ -1,5 +1,5 @@
 from cypari2.gen cimport Gen as pari_gen
-from .power_series_ring_element cimport PowerSeries
+from sage.rings.power_series_ring_element cimport PowerSeries
 
 cdef class PowerSeries_pari(PowerSeries):
     cdef pari_gen g
diff --git a/src/sage/rings/power_series_poly.pxd b/src/sage/rings/power_series_poly.pxd
index 80c441798a1..e37e1fb26cf 100644
--- a/src/sage/rings/power_series_poly.pxd
+++ b/src/sage/rings/power_series_poly.pxd
@@ -1,4 +1,4 @@
-from .power_series_ring_element cimport PowerSeries
+from sage.rings.power_series_ring_element cimport PowerSeries
 from sage.rings.polynomial.polynomial_element cimport Polynomial
 from sage.categories.action cimport Action
 
diff --git a/src/sage/rings/power_series_poly.pyx b/src/sage/rings/power_series_poly.pyx
index 519e13c1226..c651fe03d02 100644
--- a/src/sage/rings/power_series_poly.pyx
+++ b/src/sage/rings/power_series_poly.pyx
@@ -4,9 +4,9 @@ Power Series Methods
 
 The class ``PowerSeries_poly`` provides additional methods for univariate power series.
 """
-from .power_series_ring_element cimport PowerSeries
+from sage.rings.power_series_ring_element cimport PowerSeries
 from sage.structure.element cimport Element
-from .infinity import infinity
+from sage.rings.infinity import infinity
 from sage.libs.pari.all import pari_gen, PariError
 
 
diff --git a/src/sage/rings/power_series_ring.py b/src/sage/rings/power_series_ring.py
index b43306adb36..0349bfbe373 100644
--- a/src/sage/rings/power_series_ring.py
+++ b/src/sage/rings/power_series_ring.py
@@ -134,34 +134,36 @@
 
 """
 
-from . import power_series_poly
-from . import power_series_mpoly
-from .power_series_pari import PowerSeries_pari
-from . import power_series_ring_element
-
-from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
-from sage.rings.polynomial.multi_polynomial_ring_base import is_MPolynomialRing
-from .polynomial.polynomial_ring_constructor import PolynomialRing
-from . import laurent_series_ring
-from . import laurent_series_ring_element
-from . import integer
-from . import ring
-from .infinity import infinity
+import sage.categories.commutative_rings as commutative_rings
 import sage.misc.latex as latex
-from sage.structure.nonexact import Nonexact
-
 from sage.interfaces.abc import MagmaElement
-from sage.rings.fraction_field_element import FractionFieldElement
 from sage.misc.sage_eval import sage_eval
-
-from sage.structure.unique_representation import UniqueRepresentation
+from sage.rings import (
+    integer,
+    laurent_series_ring,
+    laurent_series_ring_element,
+    power_series_mpoly,
+    power_series_poly,
+    power_series_ring_element,
+    ring,
+)
+from sage.rings.fraction_field_element import FractionFieldElement
+from sage.rings.infinity import infinity
+from sage.rings.polynomial.multi_polynomial_ring_base import is_MPolynomialRing
+from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
+from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+from sage.rings.power_series_pari import PowerSeries_pari
 from sage.structure.category_object import normalize_names
-from sage.structure.element import parent, Expression
-import sage.categories.commutative_rings as commutative_rings
+from sage.structure.element import Expression, parent
+from sage.structure.nonexact import Nonexact
+from sage.structure.unique_representation import UniqueRepresentation
+
 _CommutativeRings = commutative_rings.CommutativeRings()
 import sage.categories.integral_domains as integral_domains
+
 _IntegralDomains = integral_domains.IntegralDomains()
 import sage.categories.fields as fields
+
 _Fields = fields.Fields()
 
 from sage.categories.complete_discrete_valuation import CompleteDiscreteValuationRings
@@ -535,7 +537,9 @@ def __init__(self, base_ring, name=None, default_prec=None, sparse=False,
             ValueError: default_prec (= -5) must be non-negative
 
         """
-        from sage.rings.finite_rings.finite_field_pari_ffelt import FiniteField_pari_ffelt
+        from sage.rings.finite_rings.finite_field_pari_ffelt import (
+            FiniteField_pari_ffelt,
+        )
 
         if implementation is None:
             if isinstance(base_ring, FiniteField_pari_ffelt):
diff --git a/src/sage/rings/power_series_ring_element.pyx b/src/sage/rings/power_series_ring_element.pyx
index b2ce51e09b3..8f570db774a 100644
--- a/src/sage/rings/power_series_ring_element.pyx
+++ b/src/sage/rings/power_series_ring_element.pyx
@@ -96,7 +96,7 @@ With power series the behavior is the same.
 # ****************************************************************************
 
 from cpython.object cimport Py_EQ, Py_NE
-from .infinity import infinity, is_Infinite
+from sage.rings.infinity import infinity, is_Infinite
 
 from sage.rings.rational_field import QQ
 
@@ -104,7 +104,7 @@ from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 import sage.misc.misc
 import sage.arith.all as arith
 import sage.misc.latex
-from .integer import Integer
+from sage.rings.integer import Integer
 from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
 
 from sage.categories.fields import Fields
diff --git a/src/sage/rings/qqbar.py b/src/sage/rings/qqbar.py
index 3be8051935c..fae8270e5f6 100644
--- a/src/sage/rings/qqbar.py
+++ b/src/sage/rings/qqbar.py
@@ -557,36 +557,52 @@
 import itertools
 import operator
 
-import sage.rings.ring
 import sage.rings.abc
 import sage.rings.number_field.number_field_base
-from sage.misc.fast_methods import Singleton
+import sage.rings.ring
+from sage.arith.misc import factor
+from sage.categories.action import Action
 from sage.misc.cachefunc import cached_method
+from sage.misc.fast_methods import Singleton
 from sage.misc.lazy_string import lazy_string
 from sage.misc.misc import increase_recursion_limit
-from sage.structure.coerce import parent_is_numerical, parent_is_real_numerical
-from sage.structure.sage_object import SageObject
-from sage.structure.richcmp import (richcmp, richcmp_method,
-                                    rich_to_bool, richcmp_not_equal,
-                                    op_EQ, op_NE, op_GT)
-from sage.rings.real_arb import RealBallField
-from sage.rings.real_mpfr import RR
-from sage.rings.real_mpfi import RealIntervalField, RIF, is_RealIntervalFieldElement, RealIntervalField_class
+from sage.rings import infinity
 from sage.rings.cc import CC
 from sage.rings.cif import CIF
-from sage.rings.complex_interval_field import ComplexIntervalField
 from sage.rings.complex_interval import is_ComplexIntervalFieldElement
-from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.polynomial.polynomial_element import Polynomial
+from sage.rings.complex_interval_field import ComplexIntervalField
 from sage.rings.integer_ring import ZZ
+from sage.rings.number_field.number_field import (
+    CyclotomicField,
+    GaussianField,
+    NumberField,
+)
+from sage.rings.number_field.number_field_element_quadratic import (
+    NumberFieldElement_gaussian,
+)
+from sage.rings.polynomial.polynomial_element import Polynomial
+from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 from sage.rings.rational_field import QQ
-from sage.rings.number_field.number_field import NumberField, GaussianField, CyclotomicField
-from sage.rings.number_field.number_field_element_quadratic import NumberFieldElement_gaussian
-from sage.arith.misc import factor
-from . import infinity
-from sage.categories.action import Action
-
+from sage.rings.real_arb import RealBallField
+from sage.rings.real_mpfi import (
+    RIF,
+    RealIntervalField,
+    RealIntervalField_class,
+    is_RealIntervalFieldElement,
+)
+from sage.rings.real_mpfr import RR
+from sage.structure.coerce import parent_is_numerical, parent_is_real_numerical
 from sage.structure.global_options import GlobalOptions
+from sage.structure.richcmp import (
+    op_EQ,
+    op_GT,
+    op_NE,
+    rich_to_bool,
+    richcmp,
+    richcmp_method,
+    richcmp_not_equal,
+)
+from sage.structure.sage_object import SageObject
 
 
 class AlgebraicField_common(sage.rings.abc.AlgebraicField_common):
@@ -888,8 +904,8 @@ def _factor_multivariate_polynomial(self, f, proof=True):
             1
 
         """
-        from sage.structure.factorization import Factorization
         from sage.interfaces.singular import singular
+        from sage.structure.factorization import Factorization
 
         if f.degree() == 0:
             return Factorization([], f.lc())
@@ -1973,8 +1989,8 @@ def random_element(self, poly_degree=2, *args, **kwds):
             sage: v                                 # random
             (0.4694381338921299?, -0.500000000000000? + 0.866025403784439?*I)
         """
-        from sage.rings.integer_ring import ZZ
         import sage.misc.prandom
+        from sage.rings.integer_ring import ZZ
         try:
             poly_degree = ZZ(poly_degree)
         except TypeError:
@@ -3437,7 +3453,7 @@ def __call__(self, elt):
             return elt.field_element_value()
         gen = elt.generator()
         sp = gen.super_poly(self)
-        assert (not (sp is None))
+        assert (sp is not None)
         return self._field(elt.field_element_value().polynomial()(sp))
 
 
@@ -5313,7 +5329,7 @@ def multiplicative_order(self):
             sage: QQbar(3/5 + 4/5*I).multiplicative_order()
             +Infinity
         """
-        if not (1 in CIF(self).norm()):
+        if 1 not in CIF(self).norm():
             return infinity.infinity
         if self.norm() != 1:
             return infinity.infinity
@@ -5797,7 +5813,7 @@ def multiplicative_order(self):
             sage: AA(5).sqrt().multiplicative_order()
             +Infinity
         """
-        if not (1 in RIF(self).abs()):
+        if 1 not in RIF(self).abs():
             return infinity.infinity
         if self == 1:
             return 1
@@ -7384,7 +7400,7 @@ def _complex_refine_interval(self, interval, prec):
                 # Give up and fall back on root isolation.
                 return self._complex_isolate_interval(interval, prec)
 
-            if not (zero in slope):
+            if zero not in slope:
                 new_range = center - val / slope
                 interval = interval.intersection(new_range)
 
@@ -8585,7 +8601,6 @@ def exactify(self):
             1000
             sage: sys.setrecursionlimit(old_recursion_limit)
         """
-        import sys
         with increase_recursion_limit(10):
             left = self._left
             right = self._right
@@ -8814,5 +8829,6 @@ def get_AA_golden_ratio():
 
 # Support Python's numbers abstract base class
 import numbers
+
 numbers.Real.register(AlgebraicReal)
 numbers.Complex.register(AlgebraicNumber)
diff --git a/src/sage/rings/quotient_ring.py b/src/sage/rings/quotient_ring.py
index 7dc4e0803c8..a6a0a8ce7f8 100644
--- a/src/sage/rings/quotient_ring.py
+++ b/src/sage/rings/quotient_ring.py
@@ -112,16 +112,15 @@
 # (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 # ****************************************************************************
+import sage.interfaces.abc
 import sage.misc.latex as latex
-from . import ring, ideal, quotient_ring_element
-from sage.structure.category_object import normalize_names
-from sage.structure.richcmp import richcmp_method, richcmp
 import sage.structure.parent_gens
-from sage.misc.cachefunc import cached_method
-from sage.categories.rings import Rings
 from sage.categories.commutative_rings import CommutativeRings
-
-import sage.interfaces.abc
+from sage.categories.rings import Rings
+from sage.misc.cachefunc import cached_method
+from sage.rings import ideal, quotient_ring_element, ring
+from sage.structure.category_object import normalize_names
+from sage.structure.richcmp import richcmp, richcmp_method
 
 _Rings = Rings()
 _CommRings = CommutativeRings()
@@ -302,7 +301,9 @@ def QuotientRing(R, I, names=None, **kwds):
     else:
         names = normalize_names(R.ngens(), names)
     if kwds.get('implementation') == 'pbori':
-        from sage.rings.polynomial.polynomial_ring_constructor import BooleanPolynomialRing_constructor as BooleanPolynomialRing
+        from sage.rings.polynomial.polynomial_ring_constructor import (
+            BooleanPolynomialRing_constructor as BooleanPolynomialRing,
+        )
         kwds.pop('implementation')
         return BooleanPolynomialRing(R.ngens(), names=names, **kwds)
     # workaround to silence warning from #34806
@@ -538,6 +539,7 @@ def construction(self):
             Finite Field of size 5
         """
         from sage.categories.pushout import QuotientFunctor
+
         # 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:
@@ -1006,7 +1008,9 @@ def ideal(self, *gens, **kwds):
         """
         if len(gens) == 1:
             gens = gens[0]
-        from sage.rings.polynomial.multi_polynomial_ring_base import MPolynomialRing_base
+        from sage.rings.polynomial.multi_polynomial_ring_base import (
+            MPolynomialRing_base,
+        )
         if not (isinstance(self.__R, MPolynomialRing_base) and self.__R._has_singular):
             # pass through
             return super().ideal(gens, **kwds)
@@ -1019,7 +1023,9 @@ def ideal(self, *gens, **kwds):
 
         global MPolynomialIdeal_quotient
         if MPolynomialIdeal_quotient is None:
-            from sage.rings.polynomial.multi_polynomial_ideal import MPolynomialIdeal_quotient
+            from sage.rings.polynomial.multi_polynomial_ideal import (
+                MPolynomialIdeal_quotient,
+            )
         return MPolynomialIdeal_quotient(self, gens, **kwds)
 
     def _element_constructor_(self, x, coerce=True):
@@ -1276,7 +1282,7 @@ def _singular_(self, singular=None):
 
         try:
             Q = self.__singular
-            if not (Q.parent() is singular):
+            if Q.parent() is not singular:
                 raise ValueError
             Q._check_valid()
             return Q
diff --git a/src/sage/rings/rational.pyx b/src/sage/rings/rational.pyx
index 0b398428006..63ad2b36bee 100644
--- a/src/sage/rings/rational.pyx
+++ b/src/sage/rings/rational.pyx
@@ -72,9 +72,9 @@ from sage.structure.richcmp cimport rich_to_bool_sgn
 import sage.rings.rational_field
 
 cimport sage.rings.integer as integer
-from .integer cimport Integer
+from sage.rings.integer cimport Integer
 
-from .integer_ring import ZZ
+from sage.rings.integer_ring import ZZ
 
 from sage.structure.coerce cimport is_numpy_type
 
diff --git a/src/sage/rings/rational_field.py b/src/sage/rings/rational_field.py
index 7296d227097..341c0d536d0 100644
--- a/src/sage/rings/rational_field.py
+++ b/src/sage/rings/rational_field.py
@@ -54,16 +54,17 @@
 
 """
 
-from .rational import Rational
-from .integer import Integer
+from sage.rings.integer import Integer
+from sage.rings.rational import Rational
 
 ZZ = None
 
-from sage.structure.parent_gens import ParentWithGens
-from sage.structure.sequence import Sequence
 import sage.rings.number_field.number_field_base as number_field_base
 from sage.misc.fast_methods import Singleton
 from sage.misc.superseded import deprecated_function_alias
+from sage.structure.parent_gens import ParentWithGens
+from sage.structure.sequence import Sequence
+
 
 class RationalField(Singleton, number_field_base.NumberField):
     r"""
@@ -317,6 +318,7 @@ def construction(self):
             (FractionField, Integer Ring)
         """
         from sage.categories.pushout import FractionField
+
         from . import integer_ring
         return FractionField(), integer_ring.ZZ
 
@@ -446,8 +448,8 @@ def __truediv__(self, I):
             sage: QQ / ZZ                                                               # needs sage.modules
             Q/Z
         """
-        from sage.rings.ideal import Ideal_generic
         from sage.groups.additive_abelian.qmodnz import QmodnZ
+        from sage.rings.ideal import Ideal_generic
         if I is ZZ:
             return QmodnZ(1)
         elif isinstance(I, Ideal_generic) and I.base_ring() is ZZ:
@@ -834,15 +836,15 @@ def hilbert_symbol_negative_at_S(self, S, b, check=True):
         - Simon Brandhorst, Juanita Duque, Anna Haensch, Manami Roy, Sandi Rudzinski (10-24-2017)
 
         """
+        from sage.arith.misc import hilbert_symbol, is_prime
+        from sage.matrix.constructor import matrix
+        from sage.modules.free_module import VectorSpace
         from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
         from sage.rings.padics.factory import Qp
-        from sage.modules.free_module import VectorSpace
-        from sage.matrix.constructor import matrix
         from sage.sets.primes import Primes
-        from sage.arith.misc import hilbert_symbol, is_prime
 
         # input checks
-        if not type(S) is list:
+        if type(S) is not list:
             raise TypeError("first argument must be a list or integer")
         # -1 is used for the infinite place
         infty = -1
@@ -1375,8 +1377,9 @@ def selmer_group_iterator(self, S, m, proof=True):
         """
         KSgens, ords = self.selmer_generators(S=S, m=m, proof=proof, orders=True)
         one = self.one()
-        from sage.misc.misc_c import prod
         from itertools import product
+
+        from sage.misc.misc_c import prod
         for ev in product(*[range(o) for o in ords]):
             yield prod((p**e for p,e in zip(KSgens, ev)), one)
 
@@ -1490,8 +1493,8 @@ def quadratic_defect(self, a, p, check=True):
             sage: QQ.quadratic_defect(5, 5)
             1
         """
-        from sage.rings.infinity import Infinity
         from sage.arith.misc import legendre_symbol
+        from sage.rings.infinity import Infinity
         if a not in self:
             raise TypeError(str(a) + " must be an element of " + str(self))
         if p.parent() == ZZ.ideal_monoid():
@@ -1596,8 +1599,8 @@ def _sympy_(self):
             sage: QQ._sympy_()                                                          # needs sympy
             Rationals
         """
-        from sympy import Rationals
         from sage.interfaces.sympy import sympy_init
+        from sympy import Rationals
         sympy_init()
         return Rationals
 
diff --git a/src/sage/rings/real_double_element_gsl.pxd b/src/sage/rings/real_double_element_gsl.pxd
index 39c36999ec4..1762c5f944b 100644
--- a/src/sage/rings/real_double_element_gsl.pxd
+++ b/src/sage/rings/real_double_element_gsl.pxd
@@ -1,4 +1,4 @@
-from .real_double cimport RealDoubleElement
+from sage.rings.real_double cimport RealDoubleElement
 
 
 cdef class RealDoubleElement_gsl(RealDoubleElement):
diff --git a/src/sage/rings/real_mpfi.pxd b/src/sage/rings/real_mpfi.pxd
index 32332d81e3e..f9fc7a1969c 100644
--- a/src/sage/rings/real_mpfi.pxd
+++ b/src/sage/rings/real_mpfi.pxd
@@ -5,8 +5,8 @@ from sage.rings.ring cimport Field
 cimport sage.rings.abc
 from sage.structure.element cimport RingElement
 
-from .rational cimport Rational
-from .real_mpfr cimport RealField_class
+from sage.rings.rational cimport Rational
+from sage.rings.real_mpfr cimport RealField_class
 
 cdef class RealIntervalFieldElement(RingElement)  # forward decl
 
diff --git a/src/sage/rings/real_mpfi.pyx b/src/sage/rings/real_mpfi.pyx
index 6e389b53ac2..58e3d8de737 100644
--- a/src/sage/rings/real_mpfi.pyx
+++ b/src/sage/rings/real_mpfi.pyx
@@ -269,11 +269,11 @@ from sage.structure.element cimport have_same_parent
 from sage.structure.parent cimport Parent
 from sage.structure.richcmp cimport richcmp
 
-from .convert.mpfi cimport mpfi_set_sage
-from .real_mpfr cimport RealField_class, RealNumber, RealField
-from .integer cimport Integer
-from .integer_ring import ZZ
-from .rational_field import QQ
+from sage.rings.convert.mpfi cimport mpfi_set_sage
+from sage.rings.real_mpfr cimport RealField_class, RealNumber, RealField
+from sage.rings.integer cimport Integer
+from sage.rings.integer_ring import ZZ
+from sage.rings.rational_field import QQ
 
 cimport sage.rings.abc
 
diff --git a/src/sage/rings/real_mpfr.pyx b/src/sage/rings/real_mpfr.pyx
index 4a091fd9191..9117dfcd4ce 100644
--- a/src/sage/rings/real_mpfr.pyx
+++ b/src/sage/rings/real_mpfr.pyx
@@ -148,12 +148,12 @@ from cypari2.stack cimport new_gen
 
 from sage.libs.mpmath.utils cimport mpfr_to_mpfval
 
-from .integer cimport Integer
-from .rational cimport Rational
+from sage.rings.integer cimport Integer
+from sage.rings.rational cimport Rational
 
 from sage.categories.map cimport Map
 
-from .real_double cimport RealDoubleElement
+from sage.rings.real_double cimport RealDoubleElement
 
 import sage.rings.rational_field
 
diff --git a/src/sage/rings/sum_of_squares.pyx b/src/sage/rings/sum_of_squares.pyx
index e6ec62dc7f1..04be29e0fdc 100644
--- a/src/sage/rings/sum_of_squares.pyx
+++ b/src/sage/rings/sum_of_squares.pyx
@@ -22,7 +22,7 @@ from libc.math cimport sqrt
 from cysignals.signals cimport sig_on, sig_off
 
 cimport sage.rings.integer as integer
-from . import integer
+from sage.rings import integer
 
 cdef int two_squares_c(uint_fast32_t n, uint_fast32_t res[2]) noexcept:
     r"""