next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
NCAlgebra :: basis(ZZ,NCLeftIdeal)

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

Synopsis

Description

This command returns a basis (or minimal generating set, if the ground ring is not a field), of a homogeneous left 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 = ncLeftIdeal{p,q,r}

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

o5 : NCLeftIdeal
i6 : bas = basis(3,I)

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

o6 : NCMatrix