Homomorphisms of rings#
We give a large number of examples of ring homomorphisms.
EXAMPLES:
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):
...
ValueError: relations do not all (canonically) map to 0
under map determined by images of generators
EXAMPLES:
Reduction to finite field:
sage: # needs sage.rings.finite_rings
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> = ZZ[]
sage: phi = R.hom([2], GF(5)); 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: # needs sage.rings.real_mpfr
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):
...
ValueError: relations do not all (canonically) map to 0
under map determined by images of generators
Homomorphism from one precision of field to another.
From smaller to bigger doesn’t make sense:
sage: R200 = RealField(200) # needs sage.rings.real_mpfr
sage: f = RR.hom( R200 ) # needs sage.rings.real_mpfr
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)) # needs sage.rings.real_mpfr
sage: f(2.5) # needs sage.rings.real_mpfr
2.500
sage: f(RR.pi()) # needs sage.rings.real_mpfr
3.142
Inclusion map from the reals to the complexes:
sage: # needs sage.rings.real_mpfr
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: # needs sage.libs.singular
sage: R.<x,y> = PolynomialRing(QQ)
sage: S.<a,b> = quo(R, ideal(1 + y^2))
sage: phi = S.hom([a^2, -b]); 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(); 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: # needs sage.rings.finite_rings
sage: k = GF(2)
sage: i = k.hom(GF(4, 'a'))
sage: i
Ring morphism:
From: Finite Field of size 2
To: Finite Field in a of size 2^2
Defn: 1 |--> 1
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: # needs sage.rings.finite_rings
sage: pi = ZZ.hom(k); pi
Natural morphism:
From: Integer Ring
To: Finite Field of size 2
sage: f = i * pi; f
Composite map:
From: Integer Ring
To: Finite Field in a of size 2^2
Defn: Natural morphism:
From: Integer Ring
To: Finite Field of size 2
then
Ring morphism:
From: Finite Field of size 2
To: Finite Field in a of size 2^2
Defn: 1 |--> 1
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: # needs sage.rings.padics
sage: phi = QQ.hom(Qp(3, print_mode='series'))
sage: phi
Ring 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: # needs sage.libs.pari
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 # needs sage.libs.pari
-2*sqrt2 + 3
sage: c = S.hom([1 - sqrt2]) # this is not valid # needs sage.libs.pari
Traceback (most recent call last):
...
ValueError: relations do not all (canonically) map to 0
under map determined by images of generators
Endomorphism of power series ring:
sage: R.<t> = PowerSeriesRing(QQ, default_prec=10); 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: 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, 10)
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: 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):
...
ValueError: relations do not all (canonically) map to 0
under map determined by images of generators
sage: R.hom([1])
Traceback (most recent call last):
...
ValueError: relations do not all (canonically) map to 0
under map determined by images of generators
Complex conjugation on cyclotomic fields:
sage: # needs sage.rings.number_field
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: # needs sage.rings.number_field
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
- class sage.rings.morphism.FrobeniusEndomorphism_generic#
Bases:
RingHomomorphism
A class implementing Frobenius endomorphisms on rings of prime characteristic.
- power()#
Return an integer \(n\) such that this endomorphism is the \(n\)-th power of the absolute (arithmetic) Frobenius.
EXAMPLES:
sage: # needs sage.rings.finite_rings 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:
RingMap
Homomorphism of rings.
- inverse()#
Return the inverse of this ring homomorphism if it exists.
Raises a
ZeroDivisionError
if the inverse does not exist.ALGORITHM:
By default, this computes a Gröbner basis of the ideal corresponding to the graph of the ring homomorphism.
EXAMPLES:
sage: R.<t> = QQ[] sage: f = R.hom([2*t - 1], R) sage: f.inverse() # needs sage.libs.singular Ring endomorphism of Univariate Polynomial Ring in t over Rational Field Defn: t |--> 1/2*t + 1/2
The following non-linear homomorphism is not invertible, but it induces an isomorphism on a quotient ring:
sage: # needs sage.libs.singular sage: R.<x,y,z> = QQ[] sage: f = R.hom([y*z, x*z, x*y], R) sage: f.inverse() Traceback (most recent call last): ... ZeroDivisionError: ring homomorphism not surjective sage: f.is_injective() True sage: Q.<x,y,z> = R.quotient(x*y*z - 1) sage: g = Q.hom([y*z, x*z, x*y], Q) sage: g.inverse() Ring endomorphism of Quotient of Multivariate Polynomial Ring in x, y, z over Rational Field by the ideal (x*y*z - 1) Defn: x |--> y*z y |--> x*z z |--> x*y
Homomorphisms over the integers are supported:
sage: S.<x,y> = ZZ[] sage: f = S.hom([x + 2*y, x + 3*y], S) sage: f.inverse() # needs sage.libs.singular Ring endomorphism of Multivariate Polynomial Ring in x, y over Integer Ring Defn: x |--> 3*x - 2*y y |--> -x + y sage: (f.inverse() * f).is_identity() # needs sage.libs.singular True
The following homomorphism is invertible over the rationals, but not over the integers:
sage: g = S.hom([x + y, x - y - 2], S) sage: g.inverse() # needs sage.libs.singular Traceback (most recent call last): ... ZeroDivisionError: ring homomorphism not surjective sage: R.<x,y> = QQ[x,y] sage: h = R.hom([x + y, x - y - 2], R) sage: (h.inverse() * h).is_identity() # needs sage.libs.singular True
This example by M. Nagata is a wild automorphism:
sage: R.<x,y,z> = QQ[] sage: sigma = R.hom([x - 2*y*(z*x+y^2) - z*(z*x+y^2)^2, ....: y + z*(z*x+y^2), z], R) sage: tau = sigma.inverse(); tau # needs sage.libs.singular Ring endomorphism of Multivariate Polynomial Ring in x, y, z over Rational Field Defn: x |--> -y^4*z - 2*x*y^2*z^2 - x^2*z^3 + 2*y^3 + 2*x*y*z + x y |--> -y^2*z - x*z^2 + y z |--> z sage: (tau * sigma).is_identity() # needs sage.libs.singular True
We compute the triangular automorphism that converts moments to cumulants, as well as its inverse, using the moment generating function. The choice of a term ordering can have a great impact on the computation time of a Gröbner basis, so here we choose a weighted ordering such that the images of the generators are homogeneous polynomials.
sage: d = 12 sage: T = TermOrder('wdegrevlex', [1..d]) sage: R = PolynomialRing(QQ, ['x%s' % j for j in (1..d)], order=T) sage: S.<t> = PowerSeriesRing(R) sage: egf = S([0] + list(R.gens())).ogf_to_egf().exp(prec=d+1) sage: phi = R.hom(egf.egf_to_ogf().list()[1:], R) sage: phi.im_gens()[:5] [x1, x1^2 + x2, x1^3 + 3*x1*x2 + x3, x1^4 + 6*x1^2*x2 + 3*x2^2 + 4*x1*x3 + x4, x1^5 + 10*x1^3*x2 + 15*x1*x2^2 + 10*x1^2*x3 + 10*x2*x3 + 5*x1*x4 + x5] sage: all(p.is_homogeneous() for p in phi.im_gens()) # needs sage.libs.singular True sage: phi.inverse().im_gens()[:5] # needs sage.libs.singular [x1, -x1^2 + x2, 2*x1^3 - 3*x1*x2 + x3, -6*x1^4 + 12*x1^2*x2 - 3*x2^2 - 4*x1*x3 + x4, 24*x1^5 - 60*x1^3*x2 + 30*x1*x2^2 + 20*x1^2*x3 - 10*x2*x3 - 5*x1*x4 + x5] sage: (phi.inverse() * phi).is_identity() # needs sage.libs.singular True
Automorphisms of number fields as well as Galois fields are supported:
sage: K.<zeta7> = CyclotomicField(7) # needs sage.rings.number_field sage: c = K.hom([1/zeta7]) # needs sage.rings.number_field sage: (c.inverse() * c).is_identity() # needs sage.libs.singular sage.rings.number_field True sage: F.<t> = GF(7^3) # needs sage.rings.finite_rings sage: f = F.hom(t^7, F) # needs sage.rings.finite_rings sage: (f.inverse() * f).is_identity() # needs sage.libs.singular sage.rings.finite_rings True
An isomorphism between the algebraic torus and the circle over a number field:
sage: # needs sage.libs.singular sage.rings.number_field sage: K.<i> = QuadraticField(-1) sage: A.<z,w> = K['z,w'].quotient('z*w - 1') sage: B.<x,y> = K['x,y'].quotient('x^2 + y^2 - 1') sage: f = A.hom([x + i*y, x - i*y], B) sage: g = f.inverse() sage: g.morphism_from_cover().im_gens() [1/2*z + 1/2*w, (-1/2*i)*z + (1/2*i)*w] sage: all(g(f(z)) == z for z in A.gens()) True
- inverse_image(I)#
Return the inverse image of an ideal or an element in the codomain of this ring homomorphism.
INPUT:
I
– an ideal or element in the codomain
OUTPUT:
For an ideal \(I\) in the codomain, this returns the largest ideal in the domain whose image is contained in \(I\).
Given an element \(b\) in the codomain, this returns an arbitrary element \(a\) in the domain such that
self(a) = b
if one such exists. The element \(a\) is unique if this ring homomorphism is injective.EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: S.<u,v> = QQ[] sage: f = R.hom([u^2, u*v, v^2], S) sage: I = S.ideal([u^6, u^5*v, u^4*v^2, u^3*v^3]) sage: J = f.inverse_image(I); J # needs sage.libs.singular Ideal (y^2 - x*z, x*y*z, x^2*z, x^2*y, x^3) of Multivariate Polynomial Ring in x, y, z over Rational Field sage: f(J) == I # needs sage.libs.singular True
Under the above homomorphism, there exists an inverse image for every element that only involves monomials of even degree:
sage: [f.inverse_image(p) for p in [u^2, u^4, u*v + u^3*v^3]] # needs sage.libs.singular [x, x^2, x*y*z + y] sage: f.inverse_image(u*v^2) # needs sage.libs.singular Traceback (most recent call last): ... ValueError: element u*v^2 does not have preimage
The image of the inverse image ideal can be strictly smaller than the original ideal:
sage: # needs sage.libs.singular sage.rings.number_field sage: S.<u,v> = QQ['u,v'].quotient('v^2 - 2') sage: f = QuadraticField(2).hom([v], S) sage: I = S.ideal(u + v) sage: J = f.inverse_image(I) sage: J.is_zero() True sage: f(J) < I True
Fractional ideals are not yet fully supported:
sage: # needs sage.rings.number_field sage: K.<a> = NumberField(QQ['x']('x^2+2')) sage: f = K.hom([-a], K) sage: I = K.ideal([a + 1]) sage: f.inverse_image(I) # needs sage.libs.singular Traceback (most recent call last): ... NotImplementedError: inverse image not implemented... sage: f.inverse_image(K.ideal(0)).is_zero() # needs sage.libs.singular True sage: f.inverse()(I) # needs sage.rings.padics Fractional ideal (-a + 1)
ALGORITHM:
By default, this computes a Gröbner basis of an ideal related to the graph of the ring homomorphism.
REFERENCES:
Proposition 2.5.12 [DS2009]
- is_invertible()#
Return whether this ring homomorphism is bijective.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: R.hom([y*z, x*z, x*y], R).is_invertible() # needs sage.libs.singular False sage: Q.<x,y,z> = R.quotient(x*y*z - 1) # needs sage.libs.singular sage: Q.hom([y*z, x*z, x*y], Q).is_invertible() # needs sage.libs.singular True
ALGORITHM:
By default, this requires the computation of a Gröbner basis.
- is_surjective()#
Return whether this ring homomorphism is surjective.
EXAMPLES:
sage: R.<x,y,z> = QQ[] sage: R.hom([y*z, x*z, x*y], R).is_surjective() # needs sage.libs.singular False sage: Q.<x,y,z> = R.quotient(x*y*z - 1) # needs sage.libs.singular sage: R.hom([y*z, x*z, x*y], Q).is_surjective() # needs sage.libs.singular True
ALGORITHM:
By default, this requires the computation of a Gröbner basis.
- kernel()#
Return the kernel ideal of this ring homomorphism.
EXAMPLES:
sage: A.<x,y> = QQ[] sage: B.<t> = QQ[] sage: f = A.hom([t^4, t^3 - t^2], B) sage: f.kernel() # needs sage.libs.singular Ideal (y^4 - x^3 + 4*x^2*y - 2*x*y^2 + x^2) of Multivariate Polynomial Ring in x, y over Rational Field
We express a Veronese subring of a polynomial ring as a quotient ring:
sage: A.<a,b,c,d> = QQ[] sage: B.<u,v> = QQ[] sage: f = A.hom([u^3, u^2*v, u*v^2, v^3], B) sage: f.kernel() == A.ideal(matrix.hankel([a, b, c], [d]).minors(2)) # needs sage.libs.singular True sage: Q = A.quotient(f.kernel()) # needs sage.libs.singular sage: Q.hom(f.im_gens(), B).is_injective() # needs sage.libs.singular True
The Steiner-Roman surface:
sage: R.<x,y,z> = QQ[] sage: S = R.quotient(x^2 + y^2 + z^2 - 1) sage: f = R.hom([x*y, x*z, y*z], S) # needs sage.libs.singular sage: f.kernel() # needs sage.libs.singular Ideal (x^2*y^2 + x^2*z^2 + y^2*z^2 - x*y*z) of Multivariate Polynomial Ring in x, y, z over Rational Field
- lift(x=None)#
Return a lifting map associated to this homomorphism, if it has been defined.
If
x
is notNone
, return the value of the lift morphism onx
.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() # needs sage.libs.singular sage: f.pushforward(R.ideal([x, 3*x + x*y + y^2])) # needs sage.libs.singular 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_cover#
Bases:
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) # needs sage.libs.singular sage: phi = S.cover(); phi # needs sage.libs.singular 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) # needs sage.libs.singular 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:
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
andS
, and letf
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: # needs sage.modules 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: # needs sage.modules 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]
- inverse()#
Return the inverse of this ring homomorphism if the underlying homomorphism of the base ring is invertible.
EXAMPLES:
sage: R.<x,y> = QQ[] sage: S.<a,b> = QQ[] sage: f = R.hom([a + b, a - b], S) sage: PR.<t> = R[] sage: PS = S['t'] sage: Pf = PR.hom(f, PS) sage: Pf.inverse() # needs sage.libs.singular Ring morphism: From: Univariate Polynomial Ring in t over Multivariate Polynomial Ring in a, b over Rational Field To: Univariate Polynomial Ring in t over Multivariate Polynomial Ring in x, y over Rational Field Defn: Induced from base ring by Ring morphism: From: Multivariate Polynomial Ring in a, b over Rational Field To: Multivariate Polynomial Ring in x, y over Rational Field Defn: a |--> 1/2*x + 1/2*y b |--> 1/2*x - 1/2*y sage: Pf.inverse()(Pf(x*t^2 + y*t)) # needs sage.libs.singular x*t^2 + y*t
- underlying_map()#
Return the underlying homomorphism of the base ring.
EXAMPLES:
sage: # needs sage.modules 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_fraction_field#
Bases:
RingHomomorphism
Morphisms between fraction fields.
- inverse()#
Return the inverse of this ring homomorphism if it exists.
EXAMPLES:
sage: S.<x> = QQ[] sage: f = S.hom([2*x - 1]) sage: g = f.extend_to_fraction_field() # needs sage.libs.singular sage: g.inverse() # needs sage.libs.singular Ring endomorphism of Fraction Field of Univariate Polynomial Ring in x over Rational Field Defn: x |--> 1/2*x + 1/2
- class sage.rings.morphism.RingHomomorphism_from_quotient#
Bases:
RingHomomorphism
A ring homomorphism with domain a generic quotient ring.
INPUT:
parent
– a ring homsetHom(R,S)
phi
– a ring homomorphismC --> S
, whereC
is the domain ofR.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 elementsim_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: # needs sage.libs.singular 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]) # needs sage.libs.singular Traceback (most recent call last): ... ValueError: relations do not all (canonically) map to 0 under map determined by images of generators
- 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]) # needs sage.libs.singular sage: S.hom([yy,xx]).morphism_from_cover() # needs sage.libs.singular 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:
RingHomomorphism
A ring homomorphism determined by the images of generators.
- base_map()#
Return the map on the base ring that is part of the defining data for this morphism. May return
None
if a coercion is used.EXAMPLES:
sage: # needs sage.rings.number_field sage: R.<x> = ZZ[] sage: K.<i> = NumberField(x^2 + 1) sage: cc = K.hom([-i]) sage: S.<y> = K[] sage: phi = S.hom([y^2], base_map=cc) sage: phi Ring endomorphism of Univariate Polynomial Ring in y over Number Field in i with defining polynomial x^2 + 1 Defn: y |--> y^2 with map of base ring sage: phi(y) y^2 sage: phi(i*y) -i*y^2 sage: phi.base_map() Composite map: From: Number Field in i with defining polynomial x^2 + 1 To: Univariate Polynomial Ring in y over Number Field in i with defining polynomial x^2 + 1 Defn: Ring endomorphism of Number Field in i with defining polynomial x^2 + 1 Defn: i |--> -i then Polynomial base injection morphism: From: Number Field in i with defining polynomial x^2 + 1 To: Univariate Polynomial Ring in y over Number Field in i with defining polynomial x^2 + 1
- 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_lift#
Bases:
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 \(x\) tox.lift()
.EXAMPLES:
sage: R.<x,y> = QQ[] sage: S.<xbar,ybar> = R.quo( (x^2 + y^2, y) ) # needs sage.libs.singular sage: S.lift() # needs sage.libs.singular 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 # needs sage.libs.singular False
Since github issue #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 ofsage.rings.ring.Ring
, as in the following example:sage: # needs sage.modules sage.rings.finite_rings 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]