TITLE: Noncommutative finitely generated algebras need not be noetherian
QUESTION [1 upvotes]: I would like to understand an example (of the title) given in the book "An Introduction to Noncommutative Noetherian Rings" by K. R. Goodearl, R. B. Warfield...
On page 8, Exercise 1E, an example of a noncommutative finitely generated algebra is given and we need to show it is not noetherian:

My question is, following the given hint, how does one show that each $e_i \in R$ ? That is, how to write it as a combination of $s$ and $t$?

REPLY [3 votes]: Notice that $e_1=1-st$. Can you write the others? What is $1-s^2t^2$?
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