basis(ZZ,NCIdeal) -- Returns a basis of an NCIdeal in a particular degree.

Synopsis

Usage:
bas = basis(n,I)
Function: basis
Inputs:
- n, an integer
- I, an object of class NCIdeal
Optional inputs:
- Degree => ...,
- Limit => ...,
- SourceRing => ...,
- Truncate => ...,
- Variables => ...,
Outputs:
- bas, an object of class NCMatrix

Description

This command returns a basis (or minimal generating set, if the ground ring is not a field), of a homogeneous two-sided ideal in a noncommutative ring.

i1 : A = QQ{x,y,z}

o1 = A

o1 : NCPolynomialRing

i2 : p = y*z + z*y - x^2

            2
o2 = zy+yz-x

o2 : A

i3 : q = x*z + z*x - y^2

         2
o3 = zx-y +xz

o3 : A

i4 : r = z^2 - x*y - y*x

      2
o4 = z -yx-xy

o4 : A

i5 : I = ncIdeal{p,q,r}

                             2      2      2
o5 = Two-sided ideal {zy+yz-x , zx-y +xz, z -yx-xy}

o5 : NCIdeal

i6 : bas = basis(3,I)

     |   2  2                3   2    2         2  2     2    2       3        2       2   2   2          2  2         3      3     2       2            3      3     2  2         2  2          2   2        2  2        3         |
o6 = | zx -y x+xzx  zyx+yzx-x   z x-yx -xyx  -yx +x y  -y x+xy   zxy-y +xzy  zy +yzy-yx   z y-y x-yxy  -zx +x z  -zyx-y +yxz+x   -zy +zxz+yx +xyx  -zxy+y +xyz-x   -zy +y z  zyz-zx +y x+yxy  -yx +xz -xyx  yz -y x-yxy  z -zyx-zxy |

o6 : NCMatrix