Consider we have already obtained the generating set $G={g_i}$ of a module $M$ in polynomial ring $R$, then, for a given element of the module, $x$, how to find the proper combination coefficient $a_i$, namely

$$x=a_1g_1+a_2g_2+cdots+a_mg_m .$$

I am not familiar with the ring theory either the module of syzygies.

But I am doing some practical calculations which involve the first module of syzygies.

For a given (multivariable) polynomial ring $R$, I understand that the first module of syzygies is related to the kernel of the following map

$$psi: R^4to R $$

$$c_1r_1+c_2r_2+c_3r_3+c_4r_4 mapsto r$$

where the l.h.s. of the map is defined by the module generated by a given number (say, four) polynomials $r_1,cdots,r_4$.

The generating set ${d_i}$ of the kernel of the above map can be derived through the Grobner basis of these polynomials, which can be obtained (without awareness of the details) by the *Mathematica* command

**GroebnerBasis()**. For the ideal generated by ${r_i}$, the command **PolynomialReduce()** seems to work.

For instance, I tried the follows

```
gb = GroebnerBasis({y - xz, x - yz, 1 - y^2, 1 - x^2}, {x, y, z})
PolynomialReduce(-1 + yz^2, gb, {x, y, z})
{-1 + yz^2, -1 + xz^2, -xz + y, x - yz}
{{1, 0, 0, 0}, 0}
```

But instead, I need to expand an element $x$ of the kernel in terms of the generating set ${d_i}$. It seems to be a basic task but I have no idea how it might work. Many thanks in advance!