Quotient Rings

AUTHORS:

  • William Stein

TESTS:

sage: R.<x> = PolynomialRing(ZZ)
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: S = R.quotient_ring(I);
sage: S == loads(dumps(S))
True
sage.rings.quotient_ring.QuotientRing(R, I, names=None)

Creates a quotient ring of the ring R by the ideal I. Variables are labeled by names. (If the quotient ring is a quotient of a polynomial ring.). If names isn’t given, ‘bar’ will be appended to the variable names in R.

INPUTS:

  • R - a commutative ring
  • I - an ideal of R
  • names - a list of strings to be used as names for the variables in the quotient ring R/I

OUTPUTS: R/I - the quotient ring R mod the ideal I

EXAMPLES:

Some simple quotient rings with the integers:

sage: R = QuotientRing(ZZ,7*ZZ); R
Quotient of Integer Ring by the ideal (7)
sage: R.gens()
(1,)
sage: 1*R(3); 6*R(3); 7*R(3)
3
4
0
sage: S = QuotientRing(ZZ,ZZ.ideal(8)); S
Quotient of Integer Ring by the ideal (8)
sage: 2*S(4)
0

With polynomial rings: (note that the variable name of the quotient ring can be specified as shown below)

sage: R.<xx> = QuotientRing(QQ[x], QQ[x].ideal(x^2 + 1)); R
Univariate Quotient Polynomial Ring in xx over Rational Field with modulus x^2 + 1
sage: R.gens(); R.gen()
(xx,)
xx
sage: for n in range(4): xx^n
1
xx
-1
-xx
sage: S = QuotientRing(QQ[x], QQ[x].ideal(x^2 - 2)); S
Univariate Quotient Polynomial Ring in xbar over Rational Field with
modulus x^2 - 2
sage: xbar = S.gen(); S.gen()
xbar
sage: for n in range(3): xbar^n
1
xbar
2

Sage coerces objects into ideals when possible:

sage: R = QuotientRing(QQ[x], x^2 + 1); R
Univariate Quotient Polynomial Ring in xbar over Rational Field with
modulus x^2 + 1

By Noether’s homomorphism theorems, the quotient of a quotient ring in R is just the quotient of R by the sum of the ideals. In this example, we end up modding out the ideal (x) from the ring QQ[x,y]:

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: S.<a,b> = QuotientRing(R,R.ideal(1 + y^2))
sage: T.<c,d> = QuotientRing(S,S.ideal(a))
sage: T
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x, y^2 + 1)
sage: R.gens(); S.gens(); T.gens()
(x, y)
(a, b)
(0, d)
sage: for n in range(4): d^n
1
d
-1
-d
class sage.rings.quotient_ring.QuotientRing_generic(R, I, names)

The quotient ring of R by the ideal I.

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ,'x')
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: S = R.quotient_ring(I); S
Quotient of Univariate Polynomial Ring in x over Integer Ring by the ideal (x^2 + 3*x + 4, x^2 + 1)
sage: R.<x,y> = PolynomialRing(QQ)
sage: S.<a,b> = R.quo(x^2 + y^2)
sage: a^2 + b^2 == 0
True
sage: S(0) == a^2 + b^2
True

EXAMPLE: Quotient of quotient

A quotient of a quotient is just the quotient of the original top ring by the sum of two ideals.

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: S.<a,b> = R.quo(1 + y^2)
sage: T.<c,d> = S.quo(a)
sage: T
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x, y^2 + 1)
sage: T.gens()
(0, d)
__call__(x, coerce=True)

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: S = R.quotient_ring(x^2+y^2)
sage: S(x)
xbar
sage: S(x^2 + y^2)
0
__cmp__(other)

Only quotients by the same ring and same ideal (with the same generators!!) are considered equal.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: S = R.quotient_ring(x^2 + y^2)
sage: S == R.quotient_ring(x^2 + y^2)
True

The ideals (x^2 + y^2) and (-x^2-y^2) are equal, but since the generators are different, the corresponding quotient rings are not equal:

sage: R.ideal(x^2+y^2) == R.ideal(-x^2 - y^2)
True
sage: R.quotient_ring(x^2 + y^2) == R.quotient_ring(-x^2 - y^2)
False
__init__(R, I, names)

Create the quotient ring of R by the ideal I.

INPUT:

  • R - a commutative ring
  • I - an ideal

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: R.quotient_ring(x^2 + y^2)
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
_coerce_impl(x)

Return the coercion of x into this quotient ring.

The rings that coerce into the quotient ring canonically, are:

  • this ring
  • anything that coerces into the ring of which this is the quotient

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: S.<a,b> = R.quotient(x^2 + y^2)
sage: S._coerce_(0)
0
sage: S._coerce_(2/3)
2/3
sage: S._coerce_(a^2 - b)
-b^2 - b
sage: S._coerce_(GF(7)(3))
...
TypeError: no canonical coercion of element into self
_latex_()

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ,'x')
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: R.quotient_ring(I)._latex_()
'\Bold{Z}[x]/\left(x^{2} + 3x + 4, x^{2} + 1\right)\Bold{Z}[x]'
_magma_init_(magma)

Return string that evaluates to Magma version of this quotient ring. This is called implicitly when doing conversions to Magma.

INPUT:

  • magma - a magma instance

EXAMPLE:

sage: P.<x,y> = PolynomialRing(GF(2))
sage: Q = P.quotient(sage.rings.ideal.FieldIdeal(P))
sage: magma(Q)                                         # optional - magma
Affine Algebra of rank 2 over GF(2)
Graded Reverse Lexicographical Order
Variables: x, y
Quotient relations:
[
x^2 + x,
y^2 + y
]
_repr_()

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ,'x')
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: R.quotient_ring(I)._repr_()
'Quotient of Univariate Polynomial Ring in x over Integer Ring by the ideal (x^2 + 3*x + 4, x^2 + 1)'
_singular_(singular=Singular)

Returns the Singular quotient ring of self if the base ring is coercible to Singular.

If a valid singular representation is found it is used otherwise a new ‘qring’ is created.

INPUT:

  • singular - Singular instance (default: the default Singular instance)

Note

This method also sets the current ring in Singular to self

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: S = R.quotient_ring(x^2+y^2)
sage: S._singular_()
//   characteristic : 0
//   number of vars : 2
//        block   1 : ordering dp
//                  : names    x y 
//        block   2 : ordering C
// quotient ring from ideal
_[1]=x2+y2
_singular_init_(singular=Singular)

Returns a newly created Singular quotient ring matching self if the base ring is coecable to Singular.

See self._singular_

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: S = R.quotient_ring(x^2+y^2)
sage: T = S._singular_init_()
sage: parent(S)
<class 'sage.rings.quotient_ring.QuotientRing_generic'>
sage: parent(T)
Singular
characteristic()

Return the characteristic of the quotient ring.

TODO: Not yet implemented!

EXAMPLES:

sage: Q = QuotientRing(ZZ,7*ZZ)
sage: Q.characteristic()
...
NotImplementedError
construction()

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ,'x')
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: R.quotient_ring(I).construction()
(QuotientFunctor, Univariate Polynomial Ring in x over Integer Ring)

TESTS:

sage: F, R = Integers(5).construction()
sage: F(R)
Ring of integers modulo 5
sage: F, R = GF(5).construction()
sage: F(R)
Finite Field of size 5
cover()

The covering ring homomorphism R \to R/I, equipped with a section.

EXAMPLES:

sage: R = ZZ.quo(3*ZZ)
sage: pi = R.cover()
sage: pi
Ring morphism:
  From: Integer Ring
  To:   Ring of integers modulo 3
  Defn: Natural quotient map
sage: pi(5)
2
sage: l = pi.lift()

EXAMPLES:

sage: R.<x,y>  = PolynomialRing(QQ)
sage: Q = R.quo( (x^2,y^2) )
sage: pi = Q.cover()
sage: pi(x^3+y)
ybar
sage: l = pi.lift(x+y^3)
sage: l
x
sage: l = pi.lift(); l
Set-theoretic ring morphism:
  From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2, y^2)
  To:   Multivariate Polynomial Ring in x, y over Rational Field
  Defn: Choice of lifting map
sage: l(x+y^3)
x
cover_ring()

Returns the cover ring of the quotient ring: that is, the original ring R from which we modded out an ideal, I.

TODO: PolynomialQuotientRings_field objects don’t have a cover_ring function.

EXAMPLES:

sage: Q = QuotientRing(ZZ,7*ZZ)
sage: Q.cover_ring()
Integer Ring
sage: Q = QuotientRing(QQ[x], x^2 + 1)
sage: Q.cover_ring()
...
AttributeError: 'PolynomialQuotientRing_field' object has no attribute 'cover_ring'
defining_ideal()

Returns the ideal generating this quotient ring.

EXAMPLES:

In the integers:

sage: Q = QuotientRing(ZZ,7*ZZ)
sage: Q.defining_ideal()
Principal ideal (7) of Integer Ring

An example involving a quotient of a quotient. By Noether’s homomorphism theorems, this is actually a quotient by a sum of two ideals:

sage: R.<x,y> = PolynomialRing(QQ,2)
sage: S.<a,b> = QuotientRing(R,R.ideal(1 + y^2))
sage: T.<c,d> = QuotientRing(S,S.ideal(a))
sage: S.defining_ideal()
Ideal (y^2 + 1) of Multivariate Polynomial Ring in x, y over Rational Field
sage: T.defining_ideal()
Ideal (x, y^2 + 1) of Multivariate Polynomial Ring in x, y over Rational Field
gen(i=0)

Returns the ith generator for this quotient ring.

EXAMPLES:

sage: R = QuotientRing(ZZ,7*ZZ)
sage: R.gen(0)
1
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: S.<a,b> = QuotientRing(R,R.ideal(1 + y^2))
sage: T.<c,d> = QuotientRing(S,S.ideal(a))
sage: T
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x, y^2 + 1)
sage: R.gen(0); R.gen(1)
x
y
sage: S.gen(0); S.gen(1)
a
b
sage: T.gen(0); T.gen(1)
0
d
ideal(*gens, **kwds)

Return the ideal of self with the given generators.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ)
sage: S = R.quotient_ring(x^2+y^2)
sage: S.ideal()
Ideal (0) of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
sage: S.ideal(x+y+1)
Ideal (xbar + ybar + 1) of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
is_field()

Returns True if the quotient ring is a field. Checks to see if the defining ideal is maximal.

TESTS:

Requires the is_maximal function to be implemented:

sage: Q = QuotientRing(ZZ,7*ZZ)
sage: Q.is_field()
...
NotImplementedError
is_integral_domain()

If this function returns True then self is definitely an integral domain. If it returns False, then either self is definitely not an integral domain or this function was unable to determine whether or not self is an integral domain.

Use self.defining_ideal().is_prime() to find out for sure whether this quotient ring is really not an integral domain, of if Sage is unable to determine the answer.

EXAMPLES:

sage: R = Integers(8)
sage: R.is_integral_domain()
False
sage: R.<a,b,c> = ZZ['a','b','c']
sage: I = R.ideal(a,b)
sage: Q = R.quotient_ring(I)
sage: Q.is_integral_domain()
...
NotImplementedError
lift()

Return the lifting map to the cover.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: S = R.quotient(x^2 + y^2)
sage: pi = S.cover(); pi
Ring morphism:
  From: Multivariate Polynomial Ring in x, y over Rational Field
  To:   Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
  Defn: Natural quotient map
sage: L = S.lift(); L
Set-theoretic ring morphism:
  From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
  To:   Multivariate Polynomial Ring in x, y over Rational Field
  Defn: Choice of lifting map
sage: L(S.0)
x
sage: L(S.1)
y

Note that some reduction may be applied so that the lift of a reduction need not equal the original element.

sage: z = pi(x^3 + 2*y^2); z
-xbar*ybar^2 + 2*ybar^2
sage: L(z)
-x*y^2 + 2*y^2
sage: L(z) == x^3 + 2*y^2
False
ngens()

Returns the number of generators for this quotient ring.

TODO: Note that ngens counts 0 as a generator. Does this make sense? That is, since 0 only generates itself and the fact that this is true for all rings, is there a way to “knock it off” of the generators list if a generator of some original ring is modded out?

EXAMPLES:

sage: R = QuotientRing(ZZ,7*ZZ)
sage: R.gens(); R.ngens()
(1,)
1
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: S.<a,b> = QuotientRing(R,R.ideal(1 + y^2))
sage: T.<c,d> = QuotientRing(S,S.ideal(a))
sage: T
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x, y^2 + 1)
sage: R.gens(); S.gens(); T.gens()
(x, y)
(a, b)
(0, d)
sage: R.ngens(); S.ngens(); T.ngens()
2
2
2
sage.rings.quotient_ring.is_QuotientRing(x)

Tests whether or not x inherits from QuotientRing_generic.

EXAMPLES:

sage: from sage.rings.quotient_ring import is_QuotientRing
sage: R.<x> = PolynomialRing(ZZ,'x')
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: S = R.quotient_ring(I)
sage: is_QuotientRing(S)
True
sage: is_QuotientRing(R)
False

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