### Quotient of Polynomial rings reduction not working

~~I am working with a quotient of a quotient of a polynomial ring and I want to make sure the reduction is done correctly. I was able to reduce my problem (hopefully) to the simplest version of it:
~~` `

~~ZZ.ideal(5).reduce(17)
~~R.<x>=PolynomialRing(QQ)

~~ZZ.ideal(5).reduce(5)
~~R.ideal(x^4).reduce(x^8+1)

~~ZZ['x'].ideal(x).reduce(x^5+3)
~~R.<x>=PolynomialRing(ZZ)

~~ZZ['x'].ideal(x)
~~

2

0

x^5 R.ideal(x^4).reduce(x^8+1)

` `

```
1
x^8 +
```~~3
~~

Principal ideal (x) of Univariate Polynomial Ring in x over Integer Ring
1

Why am I ~~had expected to obtain remainder of dividing x^n + 3 /n = 3, ~~not ~~x^5 + 3. What did I do wrong here? How can I ensure I am only working with elements of ZZ['x'] modulus a polynomial of my choice ? (What I actually have is ZZ['x']/(poly_1)/(poly_2), ~~getting the ~~first modular operation works fine but I cannot manage to reduce 2 * poly_2, x * poly_2 to 0 )~~

EDIT: It works when using QQ as a field, but not when using ZZ. Why?
` `

R.<x>=PolynomialRing(QQ)

R.ideal(R.gen()).reduce(2 * R.gen()+3)

result 1 in both cases?