7.10.3.2 lpIsSemiPrime

Usage:

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

Return:

boolean

Purpose:

Check whether A/<LM(G)> is semi-prime ring,
alternatively whether <LM(G)> is a semi-prime ideal in A.

Assume:

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

Theory:

A (two-sided) ideal I in the ring A is semi-prime, if for any a in A one has aAa subseteq I implies a in I.

Note:

lpIsSemiPrime works with the monomial algebra A/<LM(G)>. A positive answer holds for both A/<LM(G)> and A/<G>, while a negative answer applies only to A/<LM(G)> and not necessarily to A/<G>.

LIB "fpaprops.lib";
ring r = 0,(x1,x2),dp;
def R = freeAlgebra(r, 5);
setring R;
ideal G = x1*x2, x2*x1; // K<x1,x2>/<x1*x2,x2*x1> is semi prime
lpIsSemiPrime(G);
==> 1