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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)

o6 = | x*z*x-x*y^2+x^2*z y*z*x-y^3+y*x*z z^2*x-z*y^2+z*x*z x*z*y+x*y*z-x^3 y*z*y+y^2*z-y*x^2 z^2*y+z*y*z-z*x^2 x*z^2-x*y*x-x^2*y y*z^2-y^2*x-y*x*y z^3-z*y*x-z*x*y |

o6 : NCMatrix

Ways to use this method: