Let \(k\) be a field. Let \(N\) be a submodule of the free \(k[x,y]\)-module \(k[x,y]^ s\) for some \(s>0\). First, the author gives a partial structure theorem for Gröbner bases of \(N\). Secondly, she presents an algorithm for computing the primary decomposition of \(N\subset k[x,y]^ s\). In fact, if \({\mathfrak P}=\langle u,v\rangle\) is a maximal ideal containing \(\text{Ann}(k[x,y]^ s/N)\) and \(Q\) is the unique primary component of \(N\), then \(Q\) is written down explicitly. Finally, as an illustration, a submodule \(N\) of \(\mathbb{Q}[x,y]^ 3\) is considered and the primary modules corresponding to \(\langle x,y\rangle\), \(\langle x,y+1\rangle\), \(\langle x-1,y\rangle\) are computed.