# How to compute syzygy module of an ideal in a quotient ring?

I am trying to compute the syzygy module of an ideal generated by two polynomials `<p,q>`

modulo `I`

, where `I`

is another ideal. This means to compute a generating set `[(p1,q1),...,(ps,qs)]`

of the module `{(g,h): gp+hq is in I}`

. I know that in Sage, we can use singular command to compute syzygy module:

```
R.<x,y> = PolynomialRing(QQ, order='lex')
f=2*x^2+y
g=y
h=2*f+g
I=ideal(f,g,h)
M = I.syzygy_module();M
[ -2 -1 1]
[ -y 2*x^2 + y 0]
```

But this does not work with modulo `I`

:

```
R.<x,y> = PolynomialRing(QQ, order='lex')
S.<a,b>=R.quo(x^2+y^2)
I=ideal(a^2,b^2);I
M = I.syzygy_module();M
Ideal (-b^2, b^2) of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
Error in lines 4-4
Traceback (most recent call last):
```

Is there a way to do that?

If your generators for i are i1,...,is, you could just compute the syzygy module of <f,g,i1,...,is> and project on the first two coordinates. On the level of grobner basis computations, that's what you'd probably end up doing anyway.

That is a very smart solution. Thank you very much!