For any field $k$, I am trying to understand the ring $k[X_1,X_1^{-1},X_2,X_2^{-1}]$ and its maximal and prime ideals. I am wondering whether this might be Noetherian (I suspect it is but can't prove it).

Is it beneficial to see this ring as a localization somehow, what's the best way to see what the prime and maximal ideals of this ring are and whether or not it is Noetherian?


The ring $k[X_1,X_1^{-1},X_2,X_2^{-1}]$ is the localization of $k[X_1,X_2]$ with respect to the multiplicatively closed set $S$ generated by $X_1$ and $X_2$, i.e., $S = \{X_1^iX_2^j \mid i,j \in {\mathbb N}\}$.

The prime ideals of $k[X_1,X_1^{-1},X_2,X_2^{-1}]$ correspond to the prime ideals of the polynomial ring $k[X_1,X_2]$ that do not intersect $S$, i.e., to the prime ideals of $k[X_1,X_2]$ that do not contain any polynomial of the form $X_1^iX_2^j$. Likewise for the maximal ideals.

Also, $k[X_1,X_1^{-1},X_2,X_2^{-1}]$ being a localization of a Noetherian ring, is immediately Noetherian itself (since an ascending chain of ideals in the localization pulls back to an ascending chain of the original).


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