Let $I= \langle f_1, \ldots f_n \rangle \subset K[x_1,\ldots, x_n]$ be a homogeneous ideal and $\operatorname{deg}(f_i) \leq d$ then it has Grobner Basis where degree of each generator is less than or equal to $2(d^2/2+d)^{2^{n-2}}$. This is proved in The structure of polynomial ideals and Grobner bases. Does there exists any similar result for the Grobner Basis for the submodules of $M$ where $M$ is a free-module of rank $n'$ over the ring $K[x_1,\ldots, x_n]$