Homomorphisms of rings

Homomorphisms of rings

We give a large number of examples of ring homomorphisms.

EXAMPLE: Natural inclusion \(\ZZ \hookrightarrow \QQ\).

sage: H = Hom(ZZ, QQ)
sage: phi = H([1])
sage: phi(10)
10
sage: phi(3/1)
3
sage: phi(2/3)
Traceback (most recent call last):
...
TypeError: 2/3 fails to convert into the map's domain Integer Ring, but a `pushforward` method is not properly implemented

There is no homomorphism in the other direction:

sage: H = Hom(QQ, ZZ)
sage: H([1])
Traceback (most recent call last):
...
TypeError: images do not define a valid homomorphism

EXAMPLES:

Reduction to finite field:

sage: H = Hom(ZZ, GF(9, 'a'))
sage: phi = H([1])
sage: phi(5)
2
sage: psi = H([4])
sage: psi(5)
2

Map from single variable polynomial ring:

sage: R, x = PolynomialRing(ZZ, 'x').objgen()
sage: phi = R.hom([2], GF(5))
sage: phi
Ring morphism:
  From: Univariate Polynomial Ring in x over Integer Ring
  To:   Finite Field of size 5
  Defn: x |--> 2
sage: phi(x + 12)
4

Identity map on the real numbers:

sage: f = RR.hom([RR(1)]); f
Ring endomorphism of Real Field with 53 bits of precision
  Defn: 1.00000000000000 |--> 1.00000000000000
sage: f(2.5)
2.50000000000000
sage: f = RR.hom( [2.0] )
Traceback (most recent call last):
...
TypeError: images do not define a valid homomorphism

Homomorphism from one precision of field to another.

From smaller to bigger doesn’t make sense:

sage: R200 = RealField(200)
sage: f = RR.hom( R200 )
Traceback (most recent call last):
...
TypeError: Natural coercion morphism from Real Field with 53 bits of precision to Real Field with 200 bits of precision not defined.

From bigger to small does:

sage: f = RR.hom( RealField(15) )
sage: f(2.5)
2.500
sage: f(RR.pi())
3.142

Inclusion map from the reals to the complexes:

sage: i = RR.hom([CC(1)]); i
Ring morphism:
  From: Real Field with 53 bits of precision
  To:   Complex Field with 53 bits of precision
  Defn: 1.00000000000000 |--> 1.00000000000000
sage: i(RR('3.1'))
3.10000000000000

A map from a multivariate polynomial ring to itself:

sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: phi = R.hom([y,z,x^2]); phi
Ring endomorphism of Multivariate Polynomial Ring in x, y, z over Rational Field
  Defn: x |--> y
        y |--> z
        z |--> x^2
sage: phi(x+y+z)
x^2 + y + z

An endomorphism of a quotient of a multi-variate polynomial ring:

sage: R.<x,y> = PolynomialRing(QQ)
sage: S.<a,b> = quo(R, ideal(1 + y^2))
sage: phi = S.hom([a^2, -b])
sage: phi
Ring endomorphism of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (y^2 + 1)
  Defn: a |--> a^2
        b |--> -b
sage: phi(b)
-b
sage: phi(a^2 + b^2)
a^4 - 1

The reduction map from the integers to the integers modulo 8, viewed as a quotient ring:

sage: R = ZZ.quo(8*ZZ)
sage: pi = R.cover()
sage: pi
Ring morphism:
  From: Integer Ring
  To:   Ring of integers modulo 8
  Defn: Natural quotient map
sage: pi.domain()
Integer Ring
sage: pi.codomain()
Ring of integers modulo 8
sage: pi(10)
2
sage: pi.lift()
Set-theoretic ring morphism:
  From: Ring of integers modulo 8
  To:   Integer Ring
  Defn: Choice of lifting map
sage: pi.lift(13)
5

Inclusion of GF(2) into GF(4,'a'):

sage: k = GF(2)
sage: i = k.hom(GF(4, 'a'))
sage: i
Ring Coercion morphism:
  From: Finite Field of size 2
  To:   Finite Field in a of size 2^2
sage: i(0)
0
sage: a = i(1); a.parent()
Finite Field in a of size 2^2

We next compose the inclusion with reduction from the integers to GF(2):

sage: pi = ZZ.hom(k)
sage: pi
Ring Coercion morphism:
  From: Integer Ring
  To:   Finite Field of size 2
sage: f = i * pi
sage: f
Composite map:
  From: Integer Ring
  To:   Finite Field in a of size 2^2
  Defn:   Ring Coercion morphism:
          From: Integer Ring
          To:   Finite Field of size 2
        then
          Ring Coercion morphism:
          From: Finite Field of size 2
          To:   Finite Field in a of size 2^2
sage: a = f(5); a
1
sage: a.parent()
Finite Field in a of size 2^2

Inclusion from \(\QQ\) to the 3-adic field:

sage: phi = QQ.hom(Qp(3, print_mode = 'series'))
sage: phi
Ring Coercion morphism:
  From: Rational Field
  To:   3-adic Field with capped relative precision 20
sage: phi.codomain()
3-adic Field with capped relative precision 20
sage: phi(394)
1 + 2*3 + 3^2 + 2*3^3 + 3^4 + 3^5 + O(3^20)

An automorphism of a quotient of a univariate polynomial ring:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<sqrt2> = R.quo(x^2-2)
sage: sqrt2^2
2
sage: (3+sqrt2)^10
993054*sqrt2 + 1404491
sage: c = S.hom([-sqrt2])
sage: c(1+sqrt2)
-sqrt2 + 1

Note that Sage verifies that the morphism is valid:

sage: (1 - sqrt2)^2
-2*sqrt2 + 3
sage: c = S.hom([1-sqrt2])    # this is not valid
Traceback (most recent call last):
...
TypeError: images do not define a valid homomorphism

Endomorphism of power series ring:

sage: R.<t> = PowerSeriesRing(QQ); R
Power Series Ring in t over Rational Field
sage: f = R.hom([t^2]); f
Ring endomorphism of Power Series Ring in t over Rational Field
  Defn: t |--> t^2
sage: R.set_default_prec(10)
sage: s = 1/(1 + t); s
1 - t + t^2 - t^3 + t^4 - t^5 + t^6 - t^7 + t^8 - t^9 + O(t^10)
sage: f(s)
1 - t^2 + t^4 - t^6 + t^8 - t^10 + t^12 - t^14 + t^16 - t^18 + O(t^20)

Frobenius on a power series ring over a finite field:

sage: R.<t> = PowerSeriesRing(GF(5))
sage: f = R.hom([t^5]); f
Ring endomorphism of Power Series Ring in t over Finite Field of size 5
  Defn: t |--> t^5
sage: a = 2 + t + 3*t^2 + 4*t^3 + O(t^4)
sage: b = 1 + t + 2*t^2 + t^3 + O(t^5)
sage: f(a)
2 + t^5 + 3*t^10 + 4*t^15 + O(t^20)
sage: f(b)
1 + t^5 + 2*t^10 + t^15 + O(t^25)
sage: f(a*b)
2 + 3*t^5 + 3*t^10 + t^15 + O(t^20)
sage: f(a)*f(b)
2 + 3*t^5 + 3*t^10 + t^15 + O(t^20)

Homomorphism of Laurent series ring:

sage: R.<t> = LaurentSeriesRing(QQ)
sage: f = R.hom([t^3 + t]); f
Ring endomorphism of Laurent Series Ring in t over Rational Field
  Defn: t |--> t + t^3
sage: R.set_default_prec(10)
sage: s = 2/t^2 + 1/(1 + t); s
2*t^-2 + 1 - t + t^2 - t^3 + t^4 - t^5 + t^6 - t^7 + t^8 - t^9 + O(t^10)
sage: f(s)
2*t^-2 - 3 - t + 7*t^2 - 2*t^3 - 5*t^4 - 4*t^5 + 16*t^6 - 9*t^7 + O(t^8)
sage: f = R.hom([t^3]); f
Ring endomorphism of Laurent Series Ring in t over Rational Field
  Defn: t |--> t^3
sage: f(s)
2*t^-6 + 1 - t^3 + t^6 - t^9 + t^12 - t^15 + t^18 - t^21 + t^24 - t^27 + O(t^30)

Note that the homomorphism must result in a converging Laurent series, so the valuation of the image of the generator must be positive:

sage: R.hom([1/t])
Traceback (most recent call last):
...
TypeError: images do not define a valid homomorphism
sage: R.hom([1])
Traceback (most recent call last):
...
TypeError: images do not define a valid homomorphism

Complex conjugation on cyclotomic fields:

sage: K.<zeta7> = CyclotomicField(7)
sage: c = K.hom([1/zeta7]); c
Ring endomorphism of Cyclotomic Field of order 7 and degree 6
  Defn: zeta7 |--> -zeta7^5 - zeta7^4 - zeta7^3 - zeta7^2 - zeta7 - 1
sage: a = (1+zeta7)^5; a
zeta7^5 + 5*zeta7^4 + 10*zeta7^3 + 10*zeta7^2 + 5*zeta7 + 1
sage: c(a)
5*zeta7^5 + 5*zeta7^4 - 4*zeta7^2 - 5*zeta7 - 4
sage: c(zeta7 + 1/zeta7)       # this element is obviously fixed by inversion
-zeta7^5 - zeta7^4 - zeta7^3 - zeta7^2 - 1
sage: zeta7 + 1/zeta7
-zeta7^5 - zeta7^4 - zeta7^3 - zeta7^2 - 1

Embedding a number field into the reals:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<beta> = NumberField(x^3 - 2)
sage: alpha = RR(2)^(1/3); alpha
1.25992104989487
sage: i = K.hom([alpha],check=False); i
Ring morphism:
  From: Number Field in beta with defining polynomial x^3 - 2
  To:   Real Field with 53 bits of precision
  Defn: beta |--> 1.25992104989487
sage: i(beta)
1.25992104989487
sage: i(beta^3)
2.00000000000000
sage: i(beta^2 + 1)
2.58740105196820

An example from Jim Carlson:

sage: K = QQ # by the way :-)
sage: R.<a,b,c,d> = K[]; R
Multivariate Polynomial Ring in a, b, c, d over Rational Field
sage: S.<u> = K[]; S
Univariate Polynomial Ring in u over Rational Field
sage: f = R.hom([0,0,0,u], S); f
Ring morphism:
  From: Multivariate Polynomial Ring in a, b, c, d over Rational Field
  To:   Univariate Polynomial Ring in u over Rational Field
  Defn: a |--> 0
        b |--> 0
        c |--> 0
        d |--> u
sage: f(a+b+c+d)
u
sage: f( (a+b+c+d)^2 )
u^2

TESTS:

sage: H = Hom(ZZ, QQ)
sage: H == loads(dumps(H))
True
sage: K.<zeta7> = CyclotomicField(7)
sage: c = K.hom([1/zeta7])
sage: c == loads(dumps(c))
True
sage: R.<t> = PowerSeriesRing(GF(5))
sage: f = R.hom([t^5])
sage: f == loads(dumps(f))
True
class sage.rings.morphism.FrobeniusEndomorphism_generic

Bases: sage.rings.morphism.RingHomomorphism

A class implementing Frobenius endomorphisms on rings of prime characteristic.

power()

Return an integer \(n\) such that this endormorphism is the \(n\)-th power of the absolute (arithmetic) Frobenius.

EXAMPLES:

sage: K.<u> = PowerSeriesRing(GF(5))
sage: Frob = K.frobenius_endomorphism()
sage: Frob.power()
1
sage: (Frob^9).power()
9
class sage.rings.morphism.RingHomomorphism

Bases: sage.rings.morphism.RingMap

Homomorphism of rings.

inverse_image(I)

Return the inverse image of the ideal \(I\) under this ring homomorphism.

EXAMPLES:

This is not implemented in any generality yet:

sage: f = ZZ.hom(ZZ)
sage: f.inverse_image(ZZ.ideal(2))
Traceback (most recent call last):
...
NotImplementedError
is_injective()

Return whether or not this morphism is injective, or raise a NotImplementedError.

EXAMPLES:

Note that currently this is not implemented in most interesting cases:

sage: f = ZZ.hom(QQ)
sage: f.is_injective()
Traceback (most recent call last):
...
NotImplementedError
is_zero()

Return True if this is the zero map and False otherwise.

A ring homomorphism is considered to be 0 if and only if it sends the 1 element of the domain to the 0 element of the codomain. Since rings in Sage all have a 1 element, the zero homomorphism is only to a ring of order 1, where 1 == 0, e.g., the ring Integers(1).

EXAMPLES:

First an example of a map that is obviously nonzero:

sage: h = Hom(ZZ, QQ)
sage: f = h.natural_map()
sage: f.is_zero()
False

Next we make the zero ring as \(\ZZ/1\ZZ\):

sage: R = Integers(1)
sage: R
Ring of integers modulo 1
sage: h = Hom(ZZ, R)
sage: f = h.natural_map()
sage: f.is_zero()
True

Finally we check an example in characteristic 2:

sage: h = Hom(ZZ, GF(2))
sage: f = h.natural_map()
sage: f.is_zero()
False
lift(x=None)

Return a lifting homomorphism associated to this homomorphism, if it has been defined.

If x is not None, return the value of the lift morphism on x.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: f = R.hom([x,x])
sage: f(x+y)
2*x
sage: f.lift()
Traceback (most recent call last):
...
ValueError: no lift map defined
sage: g = R.hom(R)
sage: f._set_lift(g)
sage: f.lift() == g
True
sage: f.lift(x)
x
pushforward(I)

Returns the pushforward of the ideal \(I\) under this ring homomorphism.

EXAMPLES:

sage: R.<x,y> = QQ[]; S.<xx,yy> = R.quo([x^2,y^2]);  f = S.cover()
sage: f.pushforward(R.ideal([x,3*x+x*y+y^2]))
Ideal (xx, xx*yy + 3*xx) of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2, y^2)
class sage.rings.morphism.RingHomomorphism_coercion

Bases: sage.rings.morphism.RingHomomorphism

Initialize self.

INPUT:

  • parent – ring homset
  • check – bool (default: True)

EXAMPLES:

sage: f = ZZ.hom(QQ); f                    # indirect doctest
Ring Coercion morphism:
  From: Integer Ring
  To:   Rational Field

sage: f == loads(dumps(f))
True
class sage.rings.morphism.RingHomomorphism_cover

Bases: sage.rings.morphism.RingHomomorphism

A homomorphism induced by quotienting a ring out by an ideal.

EXAMPLES:

sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: S.<a,b> = R.quo(x^2 + y^2)
sage: phi = S.cover(); phi
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: phi(x+y)
a + b
kernel()

Return the kernel of this covering morphism, which is the ideal that was quotiented out by.

EXAMPLES:

sage: f = Zmod(6).cover()
sage: f.kernel()
Principal ideal (6) of Integer Ring
class sage.rings.morphism.RingHomomorphism_from_base

Bases: sage.rings.morphism.RingHomomorphism

A ring homomorphism determined by a ring homomorphism of the base ring.

AUTHOR:

  • Simon King (initial version, 2010-04-30)

EXAMPLES:

We define two polynomial rings and a ring homomorphism:

sage: R.<x,y> = QQ[]
sage: S.<z> = QQ[]
sage: f = R.hom([2*z,3*z],S)

Now we construct polynomial rings based on R and S, and let f act on the coefficients:

sage: PR.<t> = R[]
sage: PS = S['t']
sage: Pf = PR.hom(f,PS)
sage: Pf
Ring morphism:
  From: Univariate Polynomial Ring in t over Multivariate Polynomial Ring in x, y over Rational Field
  To:   Univariate Polynomial Ring in t over Univariate Polynomial Ring in z over Rational Field
  Defn: Induced from base ring by
        Ring morphism:
          From: Multivariate Polynomial Ring in x, y over Rational Field
          To:   Univariate Polynomial Ring in z over Rational Field
          Defn: x |--> 2*z
                y |--> 3*z
sage: p = (x - 4*y + 1/13)*t^2 + (1/2*x^2 - 1/3*y^2)*t + 2*y^2 + x
sage: Pf(p)
(-10*z + 1/13)*t^2 - z^2*t + 18*z^2 + 2*z

Similarly, we can construct the induced homomorphism on a matrix ring over our polynomial rings:

sage: MR = MatrixSpace(R,2,2)
sage: MS = MatrixSpace(S,2,2)
sage: M = MR([x^2 + 1/7*x*y - y^2, - 1/2*y^2 + 2*y + 1/6, 4*x^2 - 14*x, 1/2*y^2 + 13/4*x - 2/11*y])
sage: Mf = MR.hom(f,MS)
sage: Mf
Ring morphism:
  From: Full MatrixSpace of 2 by 2 dense matrices over Multivariate Polynomial Ring in x, y over Rational Field
  To:   Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in z over Rational Field
  Defn: Induced from base ring by
        Ring morphism:
          From: Multivariate Polynomial Ring in x, y over Rational Field
          To:   Univariate Polynomial Ring in z over Rational Field
          Defn: x |--> 2*z
                y |--> 3*z
sage: Mf(M)
[           -29/7*z^2 -9/2*z^2 + 6*z + 1/6]
[       16*z^2 - 28*z   9/2*z^2 + 131/22*z]

The construction of induced homomorphisms is recursive, and so we have:

sage: MPR = MatrixSpace(PR, 2)
sage: MPS = MatrixSpace(PS, 2)
sage: M = MPR([(- x + y)*t^2 + 58*t - 3*x^2 + x*y, (- 1/7*x*y - 1/40*x)*t^2 + (5*x^2 + y^2)*t + 2*y, (- 1/3*y + 1)*t^2 + 1/3*x*y + y^2 + 5/2*y + 1/4, (x + 6*y + 1)*t^2])
sage: MPf = MPR.hom(f,MPS); MPf
Ring morphism:
  From: Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in t over Multivariate Polynomial Ring in x, y over Rational Field
  To:   Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in t over Univariate Polynomial Ring in z over Rational Field
  Defn: Induced from base ring by
        Ring morphism:
          From: Univariate Polynomial Ring in t over Multivariate Polynomial Ring in x, y over Rational Field
          To:   Univariate Polynomial Ring in t over Univariate Polynomial Ring in z over Rational Field
          Defn: Induced from base ring by
                Ring morphism:
                  From: Multivariate Polynomial Ring in x, y over Rational Field
                  To:   Univariate Polynomial Ring in z over Rational Field
                  Defn: x |--> 2*z
                        y |--> 3*z
sage: MPf(M)
[                    z*t^2 + 58*t - 6*z^2 (-6/7*z^2 - 1/20*z)*t^2 + 29*z^2*t + 6*z]
[    (-z + 1)*t^2 + 11*z^2 + 15/2*z + 1/4                           (20*z + 1)*t^2]
underlying_map()

Return the underlying homomorphism of the base ring.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: S.<z> = QQ[]
sage: f = R.hom([2*z,3*z],S)
sage: MR = MatrixSpace(R,2)
sage: MS = MatrixSpace(S,2)
sage: g = MR.hom(f,MS)
sage: g.underlying_map() == f
True
class sage.rings.morphism.RingHomomorphism_from_quotient

Bases: sage.rings.morphism.RingHomomorphism

A ring homomorphism with domain a generic quotient ring.

INPUT:

  • parent – a ring homset Hom(R,S)
  • phi – a ring homomorphism C --> S, where C is the domain of R.cover()

OUTPUT: a ring homomorphism

The domain \(R\) is a quotient object \(C \to R\), and R.cover() is the ring homomorphism \(\varphi: C \to R\). The condition on the elements im_gens of \(S\) is that they define a homomorphism \(C \to S\) such that each generator of the kernel of \(\varphi\) maps to \(0\).

EXAMPLES:

sage: R.<x, y, z> = PolynomialRing(QQ, 3)
sage: S.<a, b, c> = R.quo(x^3 + y^3 + z^3)
sage: phi = S.hom([b, c, a]); phi
Ring endomorphism of Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field by the ideal (x^3 + y^3 + z^3)
  Defn: a |--> b
        b |--> c
        c |--> a
sage: phi(a+b+c)
a + b + c
sage: loads(dumps(phi)) == phi
True

Validity of the homomorphism is determined, when possible, and a TypeError is raised if there is no homomorphism sending the generators to the given images:

sage: S.hom([b^2, c^2, a^2])
Traceback (most recent call last):
...
TypeError: images do not define a valid homomorphism
morphism_from_cover()

Underlying morphism used to define this quotient map, i.e., the morphism from the cover of the domain.

EXAMPLES:

sage: R.<x,y> = QQ[]; S.<xx,yy> = R.quo([x^2,y^2])
sage: S.hom([yy,xx]).morphism_from_cover()
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: x |--> yy
        y |--> xx
class sage.rings.morphism.RingHomomorphism_im_gens

Bases: sage.rings.morphism.RingHomomorphism

A ring homomorphism determined by the images of generators.

im_gens()

Return the images of the generators of the domain.

OUTPUT:

  • list – a copy of the list of gens (it is safe to change this)

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: f = R.hom([x,x+y])
sage: f.im_gens()
[x, x + y]

We verify that the returned list of images of gens is a copy, so changing it doesn’t change f:

sage: f.im_gens()[0] = 5
sage: f.im_gens()
[x, x + y]
class sage.rings.morphism.RingMap

Bases: sage.categories.morphism.Morphism

Set-theoretic map between rings.

class sage.rings.morphism.RingMap_lift

Bases: sage.rings.morphism.RingMap

Given rings \(R\) and \(S\) such that for any \(x \in R\) the function x.lift() is an element that naturally coerces to \(S\), this returns the set-theoretic ring map \(R \to S\) sending \(x\) to x.lift().

EXAMPLES:

sage: R, (x,y) = PolynomialRing(QQ, 2, 'xy').objgens()
sage: S.<xbar,ybar> = R.quo( (x^2 + y^2, y) )
sage: S.lift()
Set-theoretic ring morphism:
  From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2, y)
  To:   Multivariate Polynomial Ring in x, y over Rational Field
  Defn: Choice of lifting map
sage: S.lift() == 0
False

Since trac ticket #11068, it is possible to create quotient rings of non-commutative rings by two-sided ideals. It was needed to modify RingMap_lift so that rings can be accepted that are no instances of sage.rings.ring.Ring, as in the following example:

sage: MS = MatrixSpace(GF(5),2,2)
sage: I = MS*[MS.0*MS.1,MS.2+MS.3]*MS
sage: Q = MS.quo(I)
sage: Q.0*Q.1   # indirect doctest
[0 1]
[0 0]
sage.rings.morphism.is_RingHomomorphism(phi)

Return True if phi is of type RingHomomorphism.

EXAMPLES:

sage: f = Zmod(8).cover()
sage: sage.rings.morphism.is_RingHomomorphism(f)
True
sage: sage.rings.morphism.is_RingHomomorphism(2/3)
False

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Space of homomorphisms between two rings

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