# Revision history [back]

### does sage allow computation of a groebner basis of an ideal in a polynomial ring quotiented by another an ideal ?

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1;I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

### does sage allow computation of a groebner basis of an ideal in a polynomial ring quotiented by another an ideal ?

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1;I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

### does sage allow computation of a groebner basis of an ideal in a polynomial ring quotiented by another an ideal ?

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1;I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

### does sage allow computation of a groebner basis of an ideal in a polynomial ring quotiented by another an ideal ?

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1;I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

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

from sage.rings.quotient_ring import is_QuotientRing is_QuotientRing\ F=ZZ.quo(3ZZ);F \ A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1;I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

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

from sage.rings.quotient_ring import is_QuotientRing\ is_QuotientRing

F=ZZ.quo(3ZZ);F \ A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1;I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

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

from sage.rings.quotient_ring import is_QuotientRing

F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1;I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

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

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1;I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

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

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1;I I=ideal(X^2-1,Y^2-1,Z^2-1);I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

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

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1);I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

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

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1);I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B
Below it the Sage code

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

Below it the Sage code

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1);I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;B

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

Below it is the Sage code

from sage.rings.quotient_ring import is_QuotientRing F=ZZ.quo(3ZZ);F A.<x,y,z> = PolynomialRing(F);A I=ideal(X^2-1,Y^2-1,Z^2-1);I R = A.quotient_ring(I);R x,y,z=R.gens(); J=ideal(xy+z,x+xz,xz+y);J
B = J.groebner_basis() ;Bcode .

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

Below is the Sage code .

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

Below is the Sage code .

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

Below is the Sage code .sage code

 17 None slelievre 14559 ●16 ●136 ●287 http://carva.org/samue...

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

Below is the sage codeI 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?

 18 None slelievre 14559 ●16 ●136 ●287 http://carva.org/samue...

### 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?