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Dom::Ideal -- the domains of sets of ideals

Introduction

Dom::Ideal(R) creates the domain of finitely generated ideals of the ring R.

Domain

Dom::Ideal(R)

Parameters

R - domain of category Cat::Ring

Introduction

Dom::Ideal(R)([a1, ...,an]) or Dom::Ideal(R)({a1, ...,an}) creates the ideal generated by the elements a1 through an.

Creating Elements

Dom::Ideal(R)([a1...])
Dom::Ideal(R)({a1...})

Parameters

a1... - elements of R

Categories

Cat::Monoid

Entries

coeffRing

the ring R

zero

the ideal consisting only of the zero element of R.

one

the ideal generated by R::one, i.e., R itself.

Method iszero: tests whether an ideal is zero

Method _mult: product of ideals

Method _plus: sum of ideals

Method _negate: negate an ideal

Method _subtract: difference of ideals

Method normal: normal form of an ideal

Method convert: convert list or set to ideal

Method expr: list of generators of an ideal

Example 1

We 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

Dom::BaseDomain

Changes




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