I have a problem with understanding the following solution to the problem:

A subring of a Noetherian ring is Noetherian? True or false?

Solution

let where k is a field, then A is Noetherian by Hilbert's basis theorem.

Now we want to construct a subring and then show that a sequence of ideals of this subring never becomes stationary thus the subring will not be Noetherian.

1)We start with letting where then is a subring of

as , we have then that

and

2) Important observation. If then as and

3) Hence we may construct an infinite chain of ideals

this chain never stops

as

where all

(contradition if they were contained).

Here are my questions and doubts....

Part 2) How do elements of look like?

are they polynomials generated by element and then multiplied by any element from the ring? f ex let's take then Right?

If we further assume that we can take for ex then if then and

but based on what do they conclude that ??? is it because of the def of that it is a ring all polynomials that are all divisible by ??? and these ''alone'' terms with powers of y are not divisible by

3) what does the last part of the solution say?

Could someone explain this solution to me step by step please?

Thanks