If
I and
J are both
monomial ideals, then the result will be as well. If
I and
J are both submodules of the same module, then the result will be an ideal, otherwise if
J is an ideal or ring element, then the result is a submodule containing
I.
Groebner bases will be computed as needed.
The colon operator
: may be used as an abbreviation of
quotient if no options need to be supplied.
If the second input
J is a ring element
f, then the principal ideal generated by
f is used.
The computation is not stored anywhere yet, BUT, it will soon be stored under
I.cache.QuotientComputation{J}, or
I.QuotientComputation{J}, so that the computation can be restarted after an interrupt.
i1 : R = ZZ[a,b,c];
|
i2 : F = a^3-b^2*c-11*c^2
3 2 2
o2 = a - b c - 11c
o2 : R
|
i3 : I = ideal(F,diff(a,F),diff(b,F),diff(c,F))
3 2 2 2 2
o3 = ideal (a - b c - 11c , 3a , -2b*c, - b - 22c)
o3 : Ideal of R
|
i4 : I : (ideal(a,b,c))^3
2 2
o4 = ideal (11c, 3b, 33a, b , a*b, a )
o4 : Ideal of R
|
If both arguments are submodules, the annihilator of
J/I (or
(J+I)/I) is returned.
i5 : S = QQ[x,y,z];
|
i6 : J = image vars S
o6 = image | x y z |
1
o6 : S-module, submodule of S
|
i7 : I = image symmetricPower(2,vars S)
o7 = image | x2 xy xz y2 yz z2 |
1
o7 : S-module, submodule of S
|
i8 : (I++I) : (J++J)
o8 = ideal (z, y, x)
o8 : Ideal of S
|
i9 : (I++I) : x+y+z
o9 = image | z 0 y 0 x 0 |
| 0 z 0 y 0 x |
2
o9 : S-module, submodule of S
|
i10 : quotient(I,J)
o10 = ideal (z, y, x)
o10 : Ideal of S
|
i11 : quotient(gens I, gens J)
o11 = {1} | x y z 0 0 0 |
{1} | 0 0 0 y z 0 |
{1} | 0 0 0 0 0 z |
3 6
o11 : Matrix S <--- S
|
Ideal quotients and saturations are useful for manipulating components of ideals. For example,
i12 : I = ideal(x^2-y^2, y^3)
2 2 3
o12 = ideal (x - y , y )
o12 : Ideal of S
|
i13 : J = ideal((x+y+z)^3, z^2)
3 2 2 3 2 2 2 2
o13 = ideal (x + 3x y + 3x*y + y + 3x z + 6x*y*z + 3y z + 3x*z + 3y*z +
-----------------------------------------------------------------------
3 2
z , z )
o13 : Ideal of S
|
i14 : L = intersect(I,J)
2 2 2 2 3 2 2 3 4 3 2 2
o14 = ideal (x z - y z , 2x y + 6x y + 6x*y + 2y - 3x z - 3x y*z + 3x*y z
-----------------------------------------------------------------------
3 4 2 2 3 4 3 2 2 3 3 2
+ 3y z, x - 6x y - 8x*y - 3y + 6x z + 6x y*z - 6x*y z - 6y z, y z )
o14 : Ideal of S
|
i15 : L : z^2
2 2 3
o15 = ideal (x - y , y )
o15 : Ideal of S
|
i16 : L : I == J
o16 = true
|