# polynomials – Find the combination in terms of the generating set for a specific element of the module

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!

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