graded/filtered noetherian algebras
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
where the are the filtered subspaces of A. Then we have an ascending chain of two-sided ideals (each generated by homogeneous elements) of gr(A)
that stabilizes at some n. Under these conditions we may rewrite the above chain of ideals as
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.