Merge #32606

src/sage/libs/singular/ring.pyx

@@ -31,7 +31,7 @@ from sage.libs.singular.decl cimport rDefault, GFInfo, ZnmInfo, nInitChar, AlgEx
 from sage.rings.integer cimport Integer
 from sage.rings.integer_ring cimport IntegerRing_class
 from sage.rings.integer_ring import ZZ
-from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
+import sage.rings.abc
 from sage.rings.number_field.number_field_base cimport NumberField
 from sage.rings.rational_field import RationalField
 from sage.rings.finite_rings.finite_field_base import FiniteField as FiniteField_generic
@@ -324,7 +324,7 @@ cdef ring *singular_ring_new(base_ring, n, names, term_order) except NULL:
 
         _ring = rDefault (_cf ,nvars, _names, nblcks, _order, _block0, _block1, _wvhdl)
 
-    elif is_IntegerModRing(base_ring):
+    elif isinstance(base_ring, sage.rings.abc.IntegerModRing):
 
         ch = base_ring.characteristic()
         if ch < 2:

src/sage/matrix/matrix0.pyx

@@ -44,7 +44,7 @@ from sage.categories.integral_domains import IntegralDomains
 
 from sage.rings.ring cimport CommutativeRing
 from sage.rings.ring import is_Ring
-from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
+import sage.rings.abc
 from sage.rings.integer_ring import is_IntegerRing
 
 import sage.modules.free_module
@@ -5749,7 +5749,7 @@ cdef class Matrix(sage.structure.element.Matrix):
         R = self.base_ring()
         if algorithm is None and R in _Fields:
             return ~self
-        elif algorithm is None and is_IntegerModRing(R):
+        elif algorithm is None and isinstance(R, sage.rings.abc.IntegerModRing):
             # Finite fields are handled above.
             # This is "easy" in that we either get an error or
             # the right answer. Note that of course there

src/sage/matrix/matrix2.pyx

@@ -94,7 +94,7 @@ from sage.rings.complex_double import CDF
 from sage.rings.real_mpfr import RealField
 from sage.rings.complex_mpfr import ComplexField
 from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
-from sage.misc.derivative import multi_derivative
+import sage.rings.abc
 from sage.arith.numerical_approx cimport digits_to_bits
 from copy import copy
 
@@ -822,8 +822,7 @@ cdef class Matrix(Matrix1):
         if not K.is_integral_domain():
             # The non-integral-domain case is handled almost entirely
             # separately.
-            from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
-            if is_IntegerModRing(K):
+            if isinstance(K, sage.rings.abc.IntegerModRing):
                 from sage.libs.pari import pari
                 A = pari(self.lift())
                 b = pari(B).lift()
@@ -1985,7 +1984,6 @@ cdef class Matrix(Matrix1):
             sage: A.determinant() == B.determinant()
             True
         """
-        from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
         from sage.symbolic.ring import is_SymbolicExpressionRing
 
         cdef Py_ssize_t n
@@ -2031,7 +2029,7 @@ cdef class Matrix(Matrix1):
             return d
 
         # Special case for Z/nZ or GF(p):
-        if is_IntegerModRing(R) and self.is_dense():
+        if isinstance(R, sage.rings.abc.IntegerModRing) and self.is_dense():
             import sys
             # If the characteristic is prime and smaller than a machine
             # word, use PARI.
@@ -14723,6 +14721,8 @@ cdef class Matrix(Matrix1):
             for i from 0 <= i < size:
                 PyList_Append(M,<object>f(<object>PyList_GET_ITEM(L,i)))
 
+            from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
+
             return MatrixSpace(IntegerModRing(2),
                                nrows=self._nrows,ncols=self._ncols).matrix(M)
 
@@ -15123,6 +15123,7 @@ cdef class Matrix(Matrix1):
             sage: v.derivative(x,x)
             (0, 0, 2)
         """
+        from sage.misc.derivative import multi_derivative
         return multi_derivative(self, args)
 
     def exp(self):

src/sage/matrix/matrix_rational_dense.pyx

@@ -109,7 +109,7 @@ from sage.rings.integer cimport Integer
 from sage.rings.ring import is_Ring
 from sage.rings.integer_ring import ZZ, is_IntegerRing
 from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
-from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
+import sage.rings.abc
 from sage.rings.rational_field import QQ
 from sage.arith.all import gcd
 
@@ -1487,7 +1487,7 @@ cdef class Matrix_rational_dense(Matrix_dense):
             return A
 
         from .matrix_modn_dense_double import MAX_MODULUS
-        if is_IntegerModRing(R) and R.order() < MAX_MODULUS:
+        if isinstance(R, sage.rings.abc.IntegerModRing) and R.order() < MAX_MODULUS:
             b = R.order()
             A, d = self._clear_denom()
             if not b.gcd(d).is_one():

src/sage/matrix/matrix_space.py

@@ -236,7 +236,7 @@ def get_matrix_class(R, nrows, ncols, sparse, implementation):
                     except ImportError:
                         pass
 
-            if sage.rings.finite_rings.integer_mod_ring.is_IntegerModRing(R):
+            if isinstance(R, sage.rings.abc.IntegerModRing):
                 from . import matrix_modn_dense_double, matrix_modn_dense_float
                 if R.order() < matrix_modn_dense_float.MAX_MODULUS:
                     return matrix_modn_dense_float.Matrix_modn_dense_float
@@ -326,7 +326,7 @@ def get_matrix_class(R, nrows, ncols, sparse, implementation):
     if implementation is not None:
         raise ValueError("cannot choose an implementation for sparse matrices")
 
-    if sage.rings.finite_rings.integer_mod_ring.is_IntegerModRing(R) and R.order() < matrix_modn_sparse.MAX_MODULUS:
+    if isinstance(R, sage.rings.abc.IntegerModRing) and R.order() < matrix_modn_sparse.MAX_MODULUS:
         return matrix_modn_sparse.Matrix_modn_sparse
 
     if sage.rings.rational_field.is_RationalField(R):

src/sage/modules/free_module.py

@@ -173,7 +173,7 @@
 import sage.rings.abc
 import sage.rings.integer_ring
 import sage.rings.rational_field
-import sage.rings.finite_rings.integer_mod_ring
+import sage.rings.abc
 import sage.rings.infinity
 import sage.rings.integer
 from sage.categories.principal_ideal_domains import PrincipalIdealDomains
@@ -254,7 +254,7 @@ def create_object(self, version, key):
             elif base_ring in PrincipalIdealDomains():
                 return FreeModule_ambient_pid(base_ring, rank, sparse=sparse)
 
-            elif isinstance(base_ring, sage.rings.number_field.order.Order) \
+            elif isinstance(base_ring, sage.rings.abc.Order) \
                 and base_ring.is_maximal() and base_ring.class_number() == 1:
                 return FreeModule_ambient_pid(base_ring, rank, sparse=sparse)
 
@@ -7435,7 +7435,7 @@ def element_class(R, is_sparse):
     elif sage.rings.rational_field.is_RationalField(R) and not is_sparse:
         from .vector_rational_dense import Vector_rational_dense
         return Vector_rational_dense
-    elif sage.rings.finite_rings.integer_mod_ring.is_IntegerModRing(R) and not is_sparse:
+    elif isinstance(R, sage.rings.abc.IntegerModRing) and not is_sparse:
         from .vector_mod2_dense import Vector_mod2_dense
         if R.order() == 2:
             return Vector_mod2_dense

src/sage/modules/free_module_element.pyx

@@ -123,7 +123,6 @@ from sage.rings.infinity import Infinity, AnInfinity
 from sage.rings.integer_ring import ZZ
 from sage.rings.real_double import RDF
 from sage.rings.complex_double import CDF
-from sage.misc.derivative import multi_derivative
 
 from sage.rings.ring cimport Ring
 from sage.rings.integer cimport Integer, smallInteger
@@ -3968,6 +3967,7 @@ cdef class FreeModuleElement(Vector):   # abstract base class
             sage: v.derivative(x,x)
             (0, 0, 2)
         """
+        from sage.misc.derivative import multi_derivative
         return multi_derivative(self, args)
 
     diff = derivative

src/sage/modules/vector_space_morphism.py

@@ -688,8 +688,10 @@ def linear_transformation(arg0, arg1=None, arg2=None, side='left'):
     from sage.modules.module import is_VectorSpace
     from sage.modules.free_module import VectorSpace
     from sage.categories.homset import Hom
-    from sage.symbolic.ring import SR
-    from sage.modules.vector_callable_symbolic_dense import Vector_callable_symbolic_dense
+    try:
+        from sage.modules.vector_callable_symbolic_dense import Vector_callable_symbolic_dense
+    except ImportError:
+        Vector_callable_symbolic_dense = ()
 
     if not side in ['left', 'right']:
         raise ValueError("side must be 'left' or 'right', not {0}".format(side))
@@ -734,6 +736,7 @@ def linear_transformation(arg0, arg1=None, arg2=None, side='left'):
     elif isinstance(arg2, (list, tuple)):
         pass
     elif isinstance(arg2, Vector_callable_symbolic_dense):
+        from sage.symbolic.ring import SR
         args = arg2.parent().base_ring()._arguments
         exprs = arg2.change_ring(SR)
         m = len(args)

src/sage/rings/abc.pyx

@@ -32,3 +32,18 @@ cdef class ComplexDoubleField(Field):
     """
 
     pass
+
+
+class IntegerModRing:
+    r"""
+    Abstract base class for :class:`~sage.rings.finite_rings.integer_mod_ring.IntegerModRing_generic`.
+    """
+
+    pass
+
+class Order:
+    r"""
+    Abstract base class for :class:`~sage.rings.number_field.order.Order`.
+    """
+
+    pass

src/sage/rings/finite_rings/finite_field_base.pyx

@@ -40,6 +40,7 @@ from sage.misc.persist import register_unpickle_override
 from sage.misc.cachefunc import cached_method
 from sage.misc.prandom import randrange
 from sage.rings.integer cimport Integer
+import sage.rings.abc
 from sage.misc.superseded import deprecation_cython as deprecation
 
 # Copied from sage.misc.fast_methods, used in __hash__() below.
@@ -1342,10 +1343,9 @@ cdef class FiniteField(Field):
             False
         """
         from sage.rings.integer_ring import ZZ
-        from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
         if R is int or R is long or R is ZZ:
             return True
-        if is_IntegerModRing(R) and self.characteristic().divides(R.characteristic()):
+        if isinstance(R, sage.rings.abc.IntegerModRing) and self.characteristic().divides(R.characteristic()):
             return R.hom((self.one(),), check=False)
         if is_FiniteField(R):
             if R is self:

src/sage/rings/finite_rings/integer_mod_ring.py

@@ -64,6 +64,7 @@
 
 from sage.arith.all import factor, primitive_root, CRT_basis
 import sage.rings.ring as ring
+import sage.rings.abc
 from . import integer_mod
 import sage.rings.integer as integer
 import sage.rings.integer_ring as integer_ring
@@ -243,11 +244,18 @@ def is_IntegerModRing(x):
     """
     Return ``True`` if ``x`` is an integer modulo ring.
 
+    This function is deprecated.  Use :func:`isinstance` with
+    :class:`sage.rings.abc.IntegerModRing` instead.
+
     EXAMPLES::
 
         sage: from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
         sage: R = IntegerModRing(17)
         sage: is_IntegerModRing(R)
+        doctest:warning...
+        DeprecationWarning: the function is_IntegerModRing is deprecated.
+        Use isinstance(..., sage.rings.abc.IntegerModRing) instead.
+        See https://trac.sagemath.org/32606 for details.
         True
         sage: is_IntegerModRing(GF(13))
         True
@@ -258,6 +266,9 @@ def is_IntegerModRing(x):
         sage: is_IntegerModRing(ZZ)
         False
     """
+    from sage.misc.superseded import deprecation
+    deprecation(32606, "the function is_IntegerModRing is deprecated. "
+                "Use isinstance(..., sage.rings.abc.IntegerModRing) instead.")
     return isinstance(x, IntegerModRing_generic)
 
 
@@ -297,7 +308,7 @@ def _unit_gens_primepowercase(p, r):
 
 
 @richcmp_method
-class IntegerModRing_generic(quotient_ring.QuotientRing_generic):
+class IntegerModRing_generic(quotient_ring.QuotientRing_generic, sage.rings.abc.IntegerModRing):
     """
     The ring of integers modulo `N`.
 

src/sage/rings/number_field/order.py

@@ -47,6 +47,7 @@
 from sage.rings.ring import IntegralDomain
 from sage.structure.sequence import Sequence
 from sage.rings.integer_ring import ZZ
+import sage.rings.abc
 from sage.structure.element import is_Element
 from .number_field_element import OrderElement_absolute, OrderElement_relative
 
@@ -126,7 +127,7 @@ def EquationOrder(f, names, **kwds):
     return K.order(K.gens())
 
 
-class Order(IntegralDomain):
+class Order(IntegralDomain, sage.rings.abc.Order):
     r"""
     An order in a number field.
 

src/sage/rings/polynomial/multi_polynomial_ideal.py

@@ -249,6 +249,7 @@
 from sage.misc.method_decorator import MethodDecorator
 
 from sage.rings.integer_ring import ZZ
+import sage.rings.abc
 import sage.rings.polynomial.toy_buchberger as toy_buchberger
 import sage.rings.polynomial.toy_variety as toy_variety
 import sage.rings.polynomial.toy_d_basis as toy_d_basis
@@ -4288,7 +4289,6 @@ def groebner_basis(self, algorithm='', deg_bound=None, mult_bound=None, prot=Fal
             sage: I.groebner_basis('magma:GroebnerBasis') # optional - magma
             [a + (-60)*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 + (-79/7)*c^2 + 3/7*c, c^4 + (-10/21)*c^3 + 1/84*c^2 + 1/84*c]
         """
-        from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
         from sage.rings.polynomial.multi_polynomial_sequence import PolynomialSequence
         from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
 
@@ -4331,7 +4331,7 @@ def groebner_basis(self, algorithm='', deg_bound=None, mult_bound=None, prot=Fal
                                 deg_bound=deg_bound, mult_bound=mult_bound,
                                 prot=prot, *args, **kwds)]
                     elif (R.term_order().is_global()
-                          and is_IntegerModRing(B)
+                          and isinstance(B, sage.rings.abc.IntegerModRing)
                           and not B.is_field()):
                         verbose("Warning: falling back to very slow toy implementation.", level=0)
 

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -221,14 +221,14 @@ from sage.rings.polynomial.polydict cimport ETuple
 from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
 
 # base ring imports
+import sage.rings.abc
 from sage.rings.finite_rings.finite_field_prime_modn import FiniteField_prime_modn
 from sage.rings.rational cimport Rational
 from sage.rings.rational_field import QQ
 import sage.rings.abc
 from sage.rings.integer_ring import is_IntegerRing, ZZ
 from sage.rings.integer cimport Integer
 from sage.rings.integer import GCD_list
-from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
 from sage.rings.number_field.number_field_base cimport NumberField
 
 from sage.structure.element import coerce_binop
@@ -1406,7 +1406,7 @@ cdef class MPolynomialRing_libsingular(MPolynomialRing_base):
         elif is_IntegerRing(base_ring):
             self.__singular = singular.ring("(integer)", _vars, order=order)
 
-        elif is_IntegerModRing(base_ring):
+        elif isinstance(base_ring, sage.rings.abc.IntegerModRing):
             ch = base_ring.characteristic()
             if ch.is_power_of(2):
                 exp = ch.nbits() -1

src/sage/rings/polynomial/polynomial_element.pyx

@@ -101,7 +101,6 @@ from sage.structure.element cimport (parent, have_same_parent,
 
 from sage.rings.rational_field import QQ, is_RationalField
 from sage.rings.integer_ring import ZZ, is_IntegerRing
-from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
 from sage.rings.integer cimport Integer, smallInteger
 from sage.libs.gmp.mpz cimport *
 from sage.rings.fraction_field import is_FractionField
@@ -4446,7 +4445,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
 
         n = None
 
-        if is_IntegerModRing(R) or is_IntegerRing(R):
+        if isinstance(R, sage.rings.abc.IntegerModRing) or is_IntegerRing(R):
             try:
                 G = list(self._pari_with_name().factor())
             except PariError:
@@ -8124,7 +8123,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
             if K.is_finite():
                 if multiplicities:
                     raise NotImplementedError("root finding with multiplicities for this polynomial not implemented (try the multiplicities=False option)")
-                elif is_IntegerModRing(K):
+                elif isinstance(K, sage.rings.abc.IntegerModRing):
                     # handling via the chinese remainders theorem
                     N = K.cardinality()
                     primes = N.prime_divisors()

src/sage/rings/polynomial/polynomial_ring_constructor.py

@@ -27,9 +27,9 @@
 import sage.rings.ring as ring
 import sage.rings.padics.padic_base_leaves as padic_base_leaves
 
+import sage.rings.abc
 from sage.rings.integer import Integer
 from sage.rings.finite_rings.finite_field_base import is_FiniteField
-from sage.rings.finite_rings.integer_mod_ring import is_IntegerModRing
 
 from sage.misc.cachefunc import weak_cached_function
 
@@ -694,7 +694,7 @@ def _single_variate(base_ring, name, sparse=None, implementation=None, order=Non
 
     # Specialized implementations
     specialized = None
-    if is_IntegerModRing(base_ring):
+    if isinstance(base_ring, sage.rings.abc.IntegerModRing):
         n = base_ring.order()
         if n.is_prime():
             specialized = polynomial_ring.PolynomialRing_dense_mod_p