Let $A$ be a filtered commutative algebra and $\mathrm{gr}(A)$ the associated graded algebra. Prove that if $\mathrm{gr}(A)$ is Noetherian without zero-divisors, then so is $A$.
Associated graded algebra is defined as $\mathrm{gr}(A)=\bigoplus F_i/F_{i-1}$ where $F_i$ is in filter.
Let $I $ be an ideal in $A$, then I want to construct some ideal in $\mathrm{gr}(A)$ from $I$. My idea is $I'=\bigoplus (I\cap F_i)/F_{i-1}$. I am not sure it is an ideal in $\mathrm{gr}(A)$. I don't know how to proceed.