Let $R = \mathbb Z[x,x^{-1},y,y^{-1}]$ be the ring of Laurent polynomials in variables $x$ and $y$. Let $M$ be an $R$-module generated by the infinite set $ \{a_0,b_0,a_1,b_1,...\}$.
The following relations also hold in $M$: $$ xa_0 + yb_0 = xa_1 + yb_1,\ xa_1 + yb_1 = xa_2 + yb_2, ...$$ What can I say about $M$?
Does the first relation imply $a_0 = a_1$? (I assume not...) Is $M$ free? is $M$ torsion-free? is $M$ torsionless?