$\mathbb R[X_0, X_1, ..., X_n,...]$ is polynomial a ring in infinitely many variables over $\mathbb R$.
Can someone please explain ideals to me?
How do we show that $<X_0>$ is an ideal of the ring, $<X_0,X_1>$ is an ideal of the ring, $<X_0,X_1,...X_n,...>$ is an ideal of the ring and so on are ideals of the ring?
How do we show that each of these ideals are not finitely generated?
Please show and explain.
I ask this because I have been shown that ascending chain of ideals stabilizes iff the ideals of the ring are finitely generated. With respect to the ring in this problem, I want to understand how we found the ideals of the ring and how having deals that are not finitely generated does not meet the ascending chain conditions on ideals. Eventually, I will have to prove that the ring is non-Noetherian but I seek help with these basic steps. Please help.
Each $\left<X_0,\ldots,X_n\right>$ is a finitely-generated ideal. But $I=\left<X_0,\ldots,X_n,\ldots\right>$ generated by all the $X_n$ is not finitely generated. To see this consider $I/I^2$. This is an ideal for $R/I$ where $R$ is the ring in question. But $R/I\cong \Bbb R$ and if $I$ were finitely generated over $R$ then $I/I^2$ would be a finite-dimensional vector space over $R/I$. But the images of the $X_n$ are linearly independent elements of $I/I^2$. So $I/I^2$ is an infinite-dimensional vector space, and $I$ cannot be a finitely generated ideal.