ABSTRACT
Proof. Consider the direct sum M = 0V AvA, where v runs over all words in a i , . . . , a¿ of length < k. There are a finite number (< (£+l)fc) of these, and each AvA is an ideal of A and thus f.g. as A-module, implying M is a f.g. Amodule. Let v be the vector whose ^-component is v. Then M = J2W Avw, summed over all words w in a i , . . . , a¿, and since M is Noetherian we can write M = S j = i ^#Wj for finitely many words wi,...,wt in the a¿, for suitable t. This means for any a € A that £a = ^TjVWj for suitable rj G A, and checking components we get (8.3).
