# 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?

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I think that when I use the variables x,y,z, Sage does the computation in the quotient ring R and when I use the variables X,Y,Z, Sage uses the base ring A. Therefore, I think that Sage really did the computation in R and found the Groebner basis of the ring J of R. The variables Xbar, Ybar and Zbar stand are the variables x,y,z

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