## Clarifying the question

If I understand correctly, the question is to extract
not the coefficient of the monomial `z`

(which would be
zero in this case), but the coefficient of `z`

when this is seen as a polynomial in `z`

with coefficients in polynomials in `x`

and `y`

.

In other words the expected answer here is `x*y + x`

.

Note that in Sage's symbolic ring, if a symbolic expression
is a polynomial in one of the symbolic variables, then
we can extract the coefficient as follows:

```
sage: x, y, z = SR.var('x, y, z')
sage: f = x*z + x*y*z + y + 1; f
x*y*z + x*z + y + 1
sage: f.coefficient(z, 1)
x*y + x
```

The question is how to do that when instead of a
symbolic expression in Sage's symbolic ring, our `f`

is an element in the quotient of a polynomial ring
in three variables modulo an ideal.

## Setup

We reproduce the setup from the question with
some variations, see notes below.

Define the field with three elements:

```
sage: K = GF(3); K
Finite Field of size 3
```

Define a polynomial ring in three variables:

```
sage: A.<X, Y, Z> = PolynomialRing(K); A
Multivariate Polynomial Ring in X, Y, Z
over Finite Field of size 3
```

Define an ideal in this polynomial ring:

```
sage: J = A.ideal(X^2 - 1, Y^2 - 1, Z^2 - 1)
Ideal (X^2 - 1, Y^2 - 1, Z^2 - 1)
of Multivariate Polynomial Ring in X, Y, Z
over Finite Field of size 3
```

Define the quotient ring:

```
sage: R.<x, y, z> = A.quotient(J); 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)
```

Define an element in the quotient ring:

```
sage: f = x*z + x*y*z + y + 1; f
x*y*z + x*z + y + 1
```

## Coefficient of `z`

using polynomial rings

Lift to the polynomial ring:

```
sage: F = f.lift(); F
X*Y*Z + X*Z + Y + 1
```

Define ring of polynomials in `Z`

with coefficients
in polynomials in `X`

and `Y`

over `K`

:

```
sage: B = K[('X', 'Y')]['Z']; B
Univariate Polynomial Ring in Z
over Multivariate Polynomial Ring in X, Y
over Finite Field of size 3
```

See `F`

as that kind of polynomial:

```
sage: G = B(F); G
(X*Y + X)*Z + Y + 1
```

Coefficient of `Z`

:

```
sage: D = G[1]; D
X*Y + X
```

Map back to polynomials in three variables:

```
sage: C = A(D); C
X*Y + X
```

Project down to the quotient ring:

```
sage: c = R(C); c
x*y + x
```

Note that this is an element in `R`

;
we might prefer an element in the quotient
ring $K[X, Y] / (X^2 - 1, Y^2 - 1)$.
This is left as an exercise.

## Using strings and the symbolic ring

We describe a different way to obtain the answer,
using strings and the symbolic ring.

Lift to the polynomial ring:

```
sage: F = f.lift(); F
X*Y*Z + X*Z + Y + 1
```

Extract coefficient of (the symbolic variable) `Z`

:

```
sage: C_SR = SR(str(F)).coefficient(SR('Z'), 1); C_SR
X*Y + X
```

Back to the polynomial ring:

```
sage: C = A(str(c_SR)); C
X*Y + X
```

Project to the quotient:
sage: c = R(C); c
x*y + x

## Some notes

### Integers modulo three vs the field with three elements

In Sage one gets different objects depending whether
we construct the ring $\mathbb{Z}/3\mathbb{Z}$
or the field with three elements.

Here are four ways to construct the ring

```
sage: k = Zmod(3)
sage: k = Integers(3)
sage: k = IntegerModRing(3)
sage: k = ZZ.quo(3*ZZ)
```

All give the same result:

```
sage: k
Ring of integers modulo 3
```

Here are two ways to construct the field:

```
sage: K = FiniteField(3)
sage: K = GF(3)
```

Both give the same result:

```
sage: K
Finite Field of size 3
```

Although one can ask Sage whether `k`

is a field,
and it will say yes, Sage does not consider `k`

and `K`

as equal, and the arithmetic in them is not implemented
the same way.

```
sage: k.is_field()
True
sage: k == K
False
```

When computing with a field, it is recommended
to define it using `FiniteField`

or `GF`

, since
this will take better advantage of finite field
arithmetic.

### Wishlist

While `A`

can convert a string:

```
sage: A('X*Y*Z + X*Z + Y + 1')
X*Y*Z + X*Z + Y + 1
```

it is not the case of the quotient ring `R`

:

```
sage: R('x*y*z + x*z + y + 1')
Traceback (most recent call last)
...
NameError: name 'x' is not defined
During handling of the above exception, another exception occurred:
Traceback (most recent call last)
...
TypeError: Could not find a mapping of the passed element to this ring.
```

It would be nice if one could get:

```
sage: R('x*y*z + x*z + y + 1')
x*y*z + x*z + y + 1
```

The instructions above, containing "coefficient" do not work. Sage (ver. 9.0 and the online ver. 9.1) return errors, saying that "QuotientRing_generic_with_category.element_class' object has no attribute 'coefficients'"