Suppose I have a module $M$ over a polynomial ring $R=k[X_1,\ldots,X_n]$, viewed as a submodule of $R^m$. I know the generators of $M$ $$ v_1=[a_{11},a_{12},\ldots,a_{1m}],\\ v_2=[a_{21},a_{22},\ldots,a_{2m}],\\ \vdots\\ v_r=[a_{r1},a_{r2},\ldots,a_{rm}]. $$
These generators have relations among them, e.g. $$\alpha_{11}v_1+\ldots+\alpha_{1r}v_r=0,\quad\ldots,\quad \alpha_{s1}v_1+\ldots+\alpha_{sr}v_r=0.$$
In other words, the module is generated by $v_i$'s modulo the above relations.
I would like to compute a Groebner basis for $M$.
Can someone help me with the Singular/Macaulay in doing these computations? I am stuck in "constructing" stage itself! (OR any other CAS, if it can be done easily!)
-- Mike