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I have constructed the following ideal

J={0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....}

in the ring R=F[X,Y,Z]/<x^2-1,y^2-1,z^2-1>, where x (resp; y,z) is the residue class of X (resp. Y,Z) modulo the ideal <x^2-1,y^2-1,z^2-1>. By its construction, the set J is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of J, by writing

B =J.groebner_basis(J)

but sage doesn' t know that J is the above ideal. It returns

TypeError: R must be a commutative ring,

I must copy all the elements of J and copy them in the brackets

B= J.groebner_basis(ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)), which, as I said above, is difficult.

Therefore I'd like to know whether there is another way to the task, without copying all the elements of J, and declaring the ideal of these elements, in the argument of the method groebner_basis(). Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal ot the same set ?

I have constructed the following ideal

J={0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....}

in the ring R=F[X,Y,Z]/<x^2-1,y^2-1,z^2-1>, where x (resp; y,z) is the residue class of X (resp. Y,Z) modulo the ideal <x^2-1,y^2-1,z^2-1>. By its construction, the set J is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of J, by writing

B =J.groebner_basis(J)

but sage doesn' t know that J is the above ideal. It returns

TypeError: R must be a commutative ring,

I must copy all the elements of J and copy them in the brackets

B= J.groebner_basis(ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)), which, as I said above, is difficult.

Therefore I'd like to know whether there is another way to the task, without copying all the elements of J, and declaring the ideal of these elements, in the argument of the method groebner_basis(). Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal ot the same set ?

I have constructed the following ideal

J={0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....}

in the ring R=F[X,Y,Z]/<x^2-1,y^2-1,z^2-1>, R=F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1), where x (resp; y,z) is the residue class of X (resp. Y,Z) modulo the ideal <x^2-1,y^2-1,z^2-1>. (X^2-1,Y^2-1,Z^2-1). By its construction, the set J is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of J, by writing

B =J.groebner_basis(J)

but sage doesn' t know that J is the above ideal. It returns

TypeError: R must be a commutative ring,

I must copy all the elements of J and copy them in the brackets

B= J.groebner_basis(ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)), which, as I said above, is difficult.

Therefore I'd like to know whether there is another way to the task, without copying all the elements of J, and declaring the ideal of these elements, in the argument of the method groebner_basis(). Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal ot the same set ?

I have constructed the following ideal

J={0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....}

in the ring R=F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1), where x (resp; y,z) is the residue class of X (resp. Y,Z) modulo the ideal (X^2-1,Y^2-1,Z^2-1). By its construction, the set J is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of J, by writing

B =J.groebner_basis(J)

but sage doesn' t know that J is the above ideal. It returns

TypeError: R must be a commutative ring,

I must copy all of the elements of J and copy past them in between the brackets

B= J.groebner_basis(ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)), which, as I said above, is difficult.

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of J, and declaring the ideal of these elements, in the argument of the method groebner_basis(). Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal ot the same set ?

I have constructed the following ideal

J={0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....}

in the ring R=F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1), where x (resp; y,z) is the residue class of X (resp. Y,Z) modulo the ideal (X^2-1,Y^2-1,Z^2-1). By its construction, the set J is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of J, by writing

B =J.groebner_basis(J)

but sage doesn' t know that J is the above ideal. It returns

TypeError: R must be a commutative ring,

I must copy all of the elements of J and past them between the brackets

B= J.groebner_basis(ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)), which, as I said above, is difficult.

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of J, and declaring the ideal of these elements, in the argument of the method groebner_basis(). Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal ot of the same set ?

I have constructed the following ideal

J={0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....}

in the ring R=F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1), where x (resp; y,z) is the residue class of X (resp. Y,Z) modulo the ideal (X^2-1,Y^2-1,Z^2-1). By its construction, the set J is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of J, by writing

B =J.groebner_basis(J)

but sage doesn' t know that J is the above ideal. It returns

TypeError: R must be a commutative ring,ring.

This erros seems to mean that J is an unapropriate new objet of a new Ring R !

I must copy all of the elements of J and past them between the brackets

B= J.groebner_basis(ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)), which, as I said above, is difficult.

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of J, and declaring the ideal of these elements, in the argument of the method groebner_basis(). Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set ?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....}
x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

B =J.groebner_basis(J)
= J.groebner_basis(J)

TypeError: R must be a commutative ring.
ring.

B = J.groebner_basis(ideal(0, xyx*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)),
x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....))

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(J)
J.groebner_basis()

but Sage doesn't know that $J$ is the above ideal. It returns

TypeError: R must be a commutative ring.

This error seems to mean that $J$ is an inappropriate new objet of a new ring $R$!

I must copy all of the elements of $J$ and paste them between the brackets

brackets and first write
 
H=ideal(ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....))
 
B = J.groebner_basis(ideal(0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....))

which, as H.groebner_basis()

As I said above, is difficult. it is difficult to construct the ideal H.

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, in the argument of the method groebner_basis().

Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis()

but Sage doesn't know that $J$ is the above ideal. It returns

TypeError: R must be a commutative ring.

This error seems to mean that $J$ is an inappropriate new objet of a new ring $R$!

I must copy all of the elements of $J$ and paste them between the brackets and first write

H=ideal(ideal(0, construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....))

and then compute

B = H.groebner_basis()

As I said above, it is difficult to construct the ideal H.

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis()

but Sage doesn't know that $J$ is the above ideal. It returns

TypeError: R must be a commutative ring.

This error seems to mean that $J$ is an inappropriate new objet of a new ring $R$!

I must copy all of the elements of $J$ and paste them between the brackets and first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....))

and then compute

B = H.groebner_basis()

As I said above, previously, it is difficult not easy to construct the ideal H.$H$ because this need copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis()

but Sage doesn't know that $J$ is the above ideal. It returns

TypeError: R must be a commutative ring.

This error seems to mean that $J$ is an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....))....)

and then compute

B = H.groebner_basis()

As I said previously, it is not easy to construct the ideal $H$ because this need that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis()

but Sage doesn't know that $J$ is the above ideal. It returns

TypeError: R must be a commutative ring.

This error seems to mean that $J$ is an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis()H.groebner_basis().

As I said previously, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis()

but Sage doesn't know that $J$ is the above ideal. It returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above ideal anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

As I said previously, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis()
J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above ideal anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

As I said previously, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above ideal set anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

As I said previously, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above set anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

As I said previously, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above set anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

As That works, but, as I said previously, previously said, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above set anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

That works, but, as I previously said, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is an ideal) to the ideal of the same set?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above set anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H=ideal(0, xyz - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

That works, but, as I previously said, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, Is there a way to convert a set of elements in a ring (which is already an ideal) to the underlying ideal of the same set??

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above set anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H = ideal(0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - xy - xz + yz - x + y - z, xz - yz - x + y - 1, xyz + xy + xz + yz - z - 1, xy + xz + x - y + z, -xyz - xy + xz - y*z - x - y - z, ....)
x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....)

B = H.groebner_basis().
H.groebner_basis().

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $x$ $F=GF(3) and $$x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above set anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H = ideal(0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

That works, but, as I previously said, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, is there a way to convert a set of elements in a ring (which is already an ideal) to the underlying ideal?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $F=GF(3) $F=GF(3)$ and $$x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above set anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H = ideal(0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

That works, but, as I previously said, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, is there a way to convert a set of elements in a ring (which is already an ideal) to the underlying ideal?

I have constructed the following ideal

J = {0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....}

in the ring

$$ R = F[X,Y,Z]/(X^2-1,Y^2-1,Z^2-1) $$

where $F=GF(3)$ and $$x$ $x$ (resp; $y$, $z$) is the residue class of $X$ (resp. $Y$, $Z$) modulo the ideal $(X^2-1, Y^2-1, Z^2-1)$. By its construction, the set $J$ is indeed an ideal, containing 2187 elements, so it is hard to write or copy all its elements. I want to find a Groebner basis of $J$, by writing

B = J.groebner_basis(),

but Sage returns

TypeError: R must be a commutative ring.

This error seems to mean that Sage doesn' t remember that $J$ is the above set anymore and consider $J$ as an inappropriate new objet of a new ring $R$!

I must first construct the ideal

H = ideal(0, x*y*z - x*y - x*z + y*z - x + y - z, x*z - y*z - x + y - 1, x*y*z + x*y + x*z + y*z - z - 1, x*y + x*z + x - y + z, -x*y*z - x*y + x*z - y*z - x - y - z, ....)

and then compute

B = H.groebner_basis().

That works, but, as I previously said, it is not easy to construct the ideal $H$ because that needs copy and paste 2187 elements!

Therefore I'd like to know whether there is another way to the task, without copying and pasting all of the elements of $J$, and declaring the ideal of these elements, is there a way to convert a set of elements in a ring (which is already an ideal) to the underlying ideal?