Let $ R$ be an integral domain and $ K$ is fraction field. If $ K$ is finitely generated over $ R$ then we say that $ R$ is a $ G$-domain, named after Oscar Goldman. This innocuous-looking definition is actually an extremely useful device in commutative algebra that pops up all over the place. In fact, the $ G$-domain usually comes up in the context of quotienting by a prime ideal, so let’s call a prime $ P$ in $ R$ a $ G$-ideal if $ R/P$ is a $ G$-domain. In this post, we shall see a few applications of this concept, following Kaplansky’s book Commutative Rings” for the theory and some standard examples for illustrations. At the end, we shall see a short paragraph proof of the Nullstellentsatz assuming the theory of $ G$-ideals.

Why is this concept so useful? Perhaps because of the following result: an ideal $ I$ in $ R$ is a $ G$-ideal if and only if it is the contraction of a maximal ideal in $ R[x]$ (although we won’t use it, it’s interesting to note that the nilradical is actually the intersection of all the $ G$-ideals). It’s worth looking at one direction of a proof of this result since it’s so short. First, the reader should prove that $ K$ can be generated by one element over $ R$ if and only if it is finitely generated, as an exercise.
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