  The filtrations on Zhu's algebra $A(V)$ and bimodules $A(M)$ are studied. As
an application, we prove that $A(V)$ is noetherian when $V$ is strongly
finitely generated. By using the associated graded module $\mathrm{gr} A(M)$,
we find some connections between different tensor products of $A(V)$ bimodules
$A(N)$ and $A(M)$.
