Quite in general, let M$M$ be an R$R$-module represented as the cockernel of a linear map f: G\to F$f: G\to F$ of free R-moduels F$F$ and G$G$ of rank n$n$ and m$m$. Then Sym(M)$\text{Sym}(M)$ is isomorphic (as an R-algebra) to Sym(F)/J$\text{Sym}(F)/J$ where J is the ideal of Sym(F)$\text{Sym}(F)$ generated by the image of f $f$ (the syzygies of M$M$). Choosing basis Sym(F)$\text{Sym}(F)$ is isomorphic to R[x_1,\dots,x_n]$R[x_1,\dots,x_n]$ and J$J$ is generated by m$m$ elements of degree 1 that correspond to the syzygies of M$M$.
These are well known properties of the symmetric algebra, see for example Bourbaki Algebra I: Chapters 1-3.
