How to make Sage know that a set is an ideal

edit

I have updatet the question because it contained some syntax errors

The computation

B=H.groebner_basis()

answers my preeceding question "Does sage allow computation of a groebner basis of an ideal J in the quotient ring Z/pZ[X_1,...X_r]/I?". The answer is "YES".

Please provide enough code for anyone to reproduce in a fresh Sage session. For instance what is F?

If the question is not self-contained, at least provide an explicit reference to a previous question where things are defined.

From your previous questions it would seem this is the context.

Construct a finite field, a multivariate polynomial ring over it, and the quotient by the ideal generated by three polynomials:

sage: F = GF(3)
sage: FXYZ.<X, Y, Z> = F[]
sage: R.<x, y, z> = FXYZ.quo([X^2 - 1, Y^2 - 1, Z^2 - 1])

Check what we have so far:

sage: R
Quotient of Multivariate Polynomial Ring in X, Y, Z
over Finite Field of size 3 by the ideal (X^2 - 1, Y^2 - 1, Z^2 - 1)

sage: R.gens()
(x, y, z)

sage: x^2, y^2, z^2
(1, 1, 1)

Now how do you construct the ideal J?

Thank for your remarks, the set $F$ is the finite field $GF(3)$ as you knew. The ideal $J$ is a "multicyclic code" of dimension $3$, and is equal the set of polynomials of $R$ which vanishes on a finite subset of $F^3$. Here is the code :

def code(POLS,ZER):

     C=Set({})
     for f in ENS:
         V={f.lift()(c) for c in POLS}
         if V=={0}:
            C=C+Set({f})                        
     return C

In my case, POLS is the set of polynomials of $R$ ( of degree $\leqslant (1,1)$, using the lexicographical order) and ZER a subset of $F^3$ consisting of a unique element. More time is needed to provide the codes for constructing POLS, I will provide these latter on.