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

- Function: basis
- Usage:
    `bas = basis(n,I)`
- Inputs:
  - n, an integer,
  - I, an instance of the type NCLeftIdeal,
- Optional inputs:
  - Degree => ..., default value null,
  - Limit => ..., default value infinity,
  - SourceRing => ..., default value null,
  - Strategy => ..., default value null,
  - Truncate => ..., default value false,
  - Variables => ..., default value null,
- Outputs:
  - bas, an instance of the type NCMatrix,

## 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:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NCAlgebra/NCAlgebraDoc.m2:2101:0.