# ncBasis -- Returns a basis of an noncommutative ring in specified degrees.

## Description

This command returns a basis (or minimal generating set, if the ground ring is not a field), of a graded noncommutative ring.

 i1 : A = QQ<|x,y,z|> o1 = A o1 : FreeAlgebra i2 : p = y*z + z*y - x^2 2 o2 = - x + y*z + z*y o2 : A i3 : q = x*z + z*x - y^2 2 o3 = x*z - y + z*x o3 : A i4 : r = z^2 - x*y - y*x 2 o4 = - x*y - y*x + z o4 : A i5 : I = ideal{p,q,r} 2 2 2 o5 = ideal (- x + y*z + z*y, x*z - y + z*x, - x*y - y*x + z ) o5 : Ideal of A i6 : B = A/I Using numthreads = 0 o6 = B o6 : FreeAlgebraQuotient i7 : bas = ncBasis(4,B) o7 = | y3x y4 yzyx yzy2 yzyz zy2x zy3 zyzx zyzy z2yx z2y2 z2yz z3x z3y z4 | 1 15 o7 : Matrix B <--- B

## Ways to use ncBasis :

• "ncBasis(InfiniteNumber,InfiniteNumber,Ring)"
• "ncBasis(InfiniteNumber,List,Ring)"
• "ncBasis(InfiniteNumber,ZZ,Ring)"
• "ncBasis(List,InfiniteNumber,Ring)"
• "ncBasis(List,List,Ring)"
• "ncBasis(List,Ring)"
• "ncBasis(Ring)"
• "ncBasis(ZZ,InfiniteNumber,Ring)"
• "ncBasis(ZZ,Ring)"
• "ncBasis(ZZ,ZZ,Ring)"

## For the programmer

The object ncBasis is .