Show that two modules $V_A$, $V_B$ are isomorphic
Solution 1:
Actually one can define $\phi:V_B\to V_A$ by $\phi(v)=Sv$. This map is bijective and clearly satisfies $\phi(v_1+v_2)=\phi(v_1)+\phi(v_2)$.
We have to show now that $\phi(P(X)v)=P(X)\phi(v)$. But $$\phi(P(X)v)=\phi(P(B)v)=S(P(B)v)=(SP(B))v=(P(A)S)v=P(A)(Sv)=P(X)\phi(v).$$