I am trying to answer this question:
Let $k$ be a field and $k[x,y] \cong k^{[2]}.$ Define the subring $A \subset k[x,y]$ by $A = k[x, xy, xy^2, xy^3, ...].$ Show that $A$ is not Noetherian.
And I got the following hint:
Hint: Consider the ideal $I = (x, xy, xy^2, xy^3, ...).$ Assume $xy^{n+1}= f_{0}x + f_1xy + \dots + f_n xy^n$ for $f_i \in A.$ Divide by $x$ and evaluate at $x=0.$}
But I have the following questions:
1- What are the elements of this ideal $I = (x, xy, xy^2, xy^3, ...)$ look like in general? what confuses me while trying to write an element of this ideal, is that this ideal is generated by the following infinite set $x, xy, xy^2, xy^3, ...$, could anyone help me in figuring this out please?
Let $K[x,y] = R$ for convenience. So you know that any element of $I$ is a linear combination of any of the generators, which as you rightly point out there are infinitely many of them. Nonetheless, they are very regular, so we can get some idea of what elements of this ideal look like. For example, it is clear that any element of $I$ must have degree in $x$ of at least $1$, since $x$ divides every generator. We can also see that there are no elements which have pure powers of $y$, since no such elements appear among the generators. These two facts tell you that the elements of $I$ are precisely polynomials with zero constant term and no terms composed just of powers of $y$ (including of course $0$, which is in every ideal).
To make this a bit more formal, we can write any element in $I$ as an $R$ linear combination of the generators, i.e. $$\forall f(x,y) \in I, f(x,y) = f_1(x,y)x + f_2(x,y)xy+f_3(x,y)xy^2+...+f_n(x,y)xy^n = x(f_1(x,y) + f_2(x,y)y+f_3(x,y)y^2+...+f_n(x,y)y^n)$$ where $f_1(x,y),...,f_n(x,y)$ are polynomials from the original ring $R$. In this form, one can see that there can be no non-zero constant terms and no pure powers of $y$, since each term in the sum is divisible by $x$