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

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