your algebra doesn't have to be both left and right Noetherian. only being left or right Noetherian is good enough.

so, as you mentioned, let be a filteration of and be some left ideals of such that . then

is an ascending chain of left ideals of and so there exists some positive integer such that for all

Claim. for any left ideals of of if and then

if i prove the claim, then your problem is also solved because, since for all and then by the claim we must have which proves that is left Noetherian.

Proof of the claim. suppose that then there exists (the minimal) positive integer such that note that here we have

not the reason for existance of scuh is that if for all then, since we'll get the contradiction

now let then and thus there exists some such that

hence but by the minimality of we have and so

therefore because and this is the contradiction we needed.