Module morphism

File: sage/rings/morphism.pyx (starting at line 1)

Homomorphisms of rings

We give a large number of examples of ring homomorphisms. 

EXAMPLE: Natural inclusion $\Z \hookrightarrow \Q$.
    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 must be coercible into Integer Ring

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

EXAMPLE: 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

EXAMPLE: 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

EXAMPLE: 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

EXAMPLE: 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

EXAMPLE: 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

EXAMPLE: 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

EXAMPLE: 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

EXAMPLE: 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


EXAMPLE: 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 morphism:
      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

EXAMPLE: Inclusion from $\Q$ 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)

EXAMPLE: 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

EXAMPLE: 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)

EXAMPLE: Frobenious 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)

EXAMPLE: 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
    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^-6 + 1 - t^3 + t^6 - t^9 + t^12 - t^15 + t^18 - t^21 + t^24 - t^27

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


EXAMPLE: 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

EXAMPLE: 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