Dom::Ideal -- the domains of sets
of ideals
IntroductionDom::Ideal(R) creates the domain of
finitely generated ideals of the ring R.
DomainDom::Ideal(R)
ParametersR |
- | domain of category Cat::Ring |
IntroductionDom::Ideal(R)([a1, ...,an]) or
Dom::Ideal(R)({a1, ...,an}) creates the ideal
generated by the elements a1 through an.
Creating
ElementsDom::Ideal(R)([a1...])
Dom::Ideal(R)({a1...})
Parametersa1... |
- | elements of R |
Cat::Monoid
the ring R
the ideal consisting only of the zero element of R.
the ideal generated by R::one, i.e., R
itself.
iszero(dom J)TRUE if J is the zero
ideal, and FALSE otherwise.iszero._mult(dom J1...)_mult._plus(dom J1...)_plus._negate(dom J)_negate._subtract(dom J1, dom
J2)_subtract.normal(dom J)normal.convert(list or set of ring elements
l)l.expr(dom J)J.expr.
Example
1We define R to be the polynomial ring
Q[x,y,z].
>> R:=Dom::DistributedPolynomial([x,y,z], Dom::Rational)
Dom::DistributedPolynomial([x, y, z], Dom::Rational, LexOrder)
Next, we define an ideal J over
R by giving a list of generators.
>> J:=Dom::Ideal(R)([x*y+y^2*x+x*y+z+1, z^2-x*z-y*x-7])
<[x*y^2 + 2*x*y + z + 1, - x*y - x*z + z^2 - 7]>
Since R is a polynomial ring over a field,
a Gröbner base of J can be obtained as follows:
>> normal(J)
<[z - 13*y + y*z - 7*y^2 + z^2 + 2*y*z^2 + y^2*z^2, 8*z - 7*y \
- 2*x*z + 2*z^2 - z^3 + x*z^2 + y*z^2 - 13, x*y + x*z - z^2 + \
7]>
Super-Domain