Singular

7.10.3.1 lpNoetherian

Usage:

lpNoetherian(G); G an ideal in a Letterplace ring

Return:

int
0 not Noetherian
1 left Noetherian
2 right Noetherian
3 Noetherian
4 weak Noetherian

Purpose:

Check whether the monomial algebra A/<LM(G)> is (left/right) noetherian

Assume:

- basering is a Letterplace ring
- G is a Groebner basis

Theory:

lpNoetherian works with the monomial algebra A/<LM(G)>. If it gives an affirmative answer for one of the properties, then it holds for both A/<LM(G)> and A/<G>. However, a negative answer applies only to A/<LM(G)> and not necessarily to A/<G>.

Note:

Weak Noetherian means that two-sided ideals in A/<LM(G)> satisfy the acc (ascending chain condition).

LIB "fpaprops.lib";
ring r = 0,(x,y),dp;
def R = freeAlgebra(r, 5);
setring R;
ideal G = x*x, y*x; // K<x,y>/<xx,yx> is right noetherian
lpNoetherian(G);
==> 2