Hi, I'm having a bit of bother proving the following statement:

Let A be a filtered algebra and gr(A) the associated graded algebra. If gr(A) is noetherian without zero-divisors, then so is A.

Here is my attempt:

Let A be a filtered algebra with gr(A) noetherian. We have

$\displaystyle gr(A)= \bigoplus_i S_i = \bigoplus_i {F_i(A)}/{F_{i-1}(A)}$

where the $\displaystyle F_i(A)$ are the filtered subspaces of A. Then we have an ascending chain of two-sided ideals (each generated by homogeneous elements) of gr(A)

$\displaystyle I_1 \subset \cdots \subset I_n $

that stabilizes at some n. Under these conditions we may rewrite the above chain of ideals as

$\displaystyle \bigoplus_i I_1 \cap S_i \subset \cdots \subset \bigoplus_i I_n \cap S_i$

I want to show that this new chain is a chain of ideals in the filtered algebra A thus proving A is noetherian. I don't know if this is the right way to go about this. Any suggestions or pointers would be appreciated.

Sorry if the maths doesn't come out well - this is the first time I've attempted it on this site.