### polynomial ring over another polynomial ring

I'd like to see what one multivariate polynomial ~~f ~~$f$ looks ~~like ~~like
when another one ~~g ~~$g$ vanishes, by rewriting the ~~former ~~former
as ~~f ~~$f = r + q ~~g ~~g$ and setting ~~g = 0. ~~$g = 0$. In Maple, one could do ~~that ~~that
just by computing

`r = prem( f, g, x `~~) ~~)

where ~~x ~~`x`

is one of ~~the ~~the
indeterminates of ~~f ~~`f`

and ~~g, ~~`g`

, which is effectively treating ~~f ~~`f`

and ~~g ~~`g`

as univariate polynomials over the ring of ~~polynomials ~~polynomials
in the remaining indeterminates. Unfortunately when I try to ~~do ~~do
that in Sage it gives me the following error (but perhaps ~~there's ~~there's
a simpler ~~way?):~~way?).

~~R.<y,z> = QQ["y,z"]~~

Input:

```
R.<y, z> = QQ['y, z']
S.<x> = PolynomialRing(R,
```~~"x" )~~

'x')
a = ~~x^2~~*y **x^2*y - x - *~~z~~

a.reduce(Ideal([xy-z]))

z
a.reduce(Ideal([x*y-z]))

Error:

~~ ~~

`Error in lines `~~4-4~~

4-4
Traceback (most recent call ~~last):~~

File "/cocalc/lib/python3.8/site-packages/smc_sagews/sage_server.py", last):
File "/cocalc/lib/python3.8/site-packages/smc_sagews/sage_server.py",
line 1230, in ~~execute~~

exec(

execute
exec(
File "", line 1, in ~~<module>~~

<module>
File "sage/structure/element.pyx", line ~~493, in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4703)~~

493,
in sage.structure.element.Element.__getattr__
(build/cythonized/sage/structure/element.c:4703)
return ~~self.getattr_from_category(name)~~

self.getattr_from_category(name)
File "sage/structure/element.pyx", line ~~506, in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4815)~~

506,
in sage.structure.element.Element.getattr_from_category
(build/cythonized/sage/structure/element.c:4815)
return getattr_from_other_class(self, cls, ~~name)~~

name)
File "sage/cpython/getattr.pyx", line ~~372, in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2620)~~

372,
in sage.cpython.getattr.getattr_from_other_class
(build/cythonized/sage/cpython/getattr.c:2620)
raise ~~AttributeError(dummy_error_message)~~

AttributeError: 'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense' AttributeError(dummy_error_message)
AttributeError:
'sage.rings.polynomial.polynomial_element.Polynomial_generic_dense'
object has no attribute ~~'reduce'~~

```
'reduce'
```