# ApCoCoA-1:NC.LWIdeal

From ApCoCoAWiki

## NC.LTIdeal

Leading word ideal of a finitely generated two-sided ideal in a non-commutative polynomial ring.

### Syntax

### Description

*Proposition:* Let `I` be a finitely generated two-sided ideal in a non-commutative polynomial ring `K<x[1],...,x[n]>`, and let `Ordering` be a word ordering on `<x[1],...,x[n]>`. If `G` is a Groebner basis of `I` with respect to `Ordering`. Then the leading word set `LW{G}:={LW(g): g in G}` is a generating system of the leading word ideal `LW(I)` with respect to `Ordering`.

#### Example

Use QQ[x,y,z,t]; NC.SetOrdering(<quotes>LLEX</quotes>); F1 := [[x^2], [-y,x]]; F2 := [[x,y], [-t,y]]; F3 := [[x,t], [-t,x]]; F4 := [[y,t], [-t,y]]; G := [F1,F2,F3,F4]; GB:=NC.GB(G); [NC.LW(E) | E In GB]; -- the leading word ideal of <G> w.r.t. the length-lexicographic word ordering [[y, t], [x, t], [x, y], [x^2], [t, y^2], [y^2, x]] -------------------------------

### See also