ncGraphIdeal -- Compute the graph ideal of a ring map between noncommutative rings.

Synopsis

Usage:

I = ncGraphIdeal f
Inputs:
- f, a ring map,
Outputs:
- I, an ideal,

Description

This function creates the graph ideal of a ring map between noncommutative rings. It creates the free product of the source and target, and forms the ideal generated by $v - f(v)$ for all variables $v$ in the source.

i1 : A = QQ<|a,b,c|>

o1 = A

o1 : FreeAlgebra

i2 : B = QQ<|x,y|>

o2 = B

o2 : FreeAlgebra

i3 : f = map(B,A,{x*y*x,y*x*y,x*y})

o3 = map (B, A, {x*y*x, y*x*y, x*y})

o3 : RingMap B <-- A

i4 : I = ncGraphIdeal f

o4 = ideal (- x*y*x + a, - y*x*y + b, - x*y + c)

o4 : Ideal of QQ <|x, y, a, b, c|>

i5 : Igb = NCGB(I,10)
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o5 = | xy-c cx-a yc-b |

                                  1                           3
o5 : Matrix (QQ <|x, y, a, b, c|>)  <-- (QQ <|x, y, a, b, c|>)

Those generators of the Groebner basis that involve only the variables in the domain are a Groebner basis of the kernel of the ring map.

Ways to use ncGraphIdeal :

ncGraphIdeal(RingMap)

For the programmer

The object ncGraphIdeal is a method function.