# does sage allow computation of a groebner basis of an ideal J in the quotient ring Z/pZ[X_1,...X_r]/I?

I define the following rings and ideals:

```
sage: F = ZZ.quo(3*ZZ); F
Ring of integers modulo 3
sage: A.<X, Y, Z> = PolynomialRing(F); A
Multivariate Polynomial Ring in X, Y, Z
over Ring of integers modulo 3
sage: I = ideal(X^2 - 1, Y^2 - 1, Z^2 - 1); I
Ideal (X^2 + 2, Y^2 + 2, Z^2 + 2)
of Multivariate Polynomial Ring in X, Y, Z
over Ring of integers modulo 3
sage: R = A.quotient_ring(I); R
Quotient of Multivariate Polynomial Ring in X, Y, Z
over Ring of integers modulo 3
by the ideal (X^2 + 2, Y^2 + 2, Z^2 + 2)
sage: x, y, z = R.gens()
sage: J = ideal(x*y + z, x + x*z, x*z + y); J
Ideal (Xbar*Ybar + Zbar, Xbar*Zbar + Xbar, Xbar*Zbar + Ybar)
of Quotient of Multivariate Polynomial Ring in X, Y, Z
over Ring of integers modulo 3
by the ideal (X^2 + 2, Y^2 + 2, Z^2 + 2)
```

I then compute:

```
sage: B = J.groebner_basis()
```

and examine the result of that computation:

```
sage: B
[Xbar + 2*Ybar, Zbar + 1]
```

Is it really the Groebner basis of `J`

?

Did Sage really compute in A/I and not in A?