Kaplansky defines a G domain as an integral domain whose field of fractions is generated as a ring over the domain by finitely many (and hence one) elements. He gives some interesting equivalent conditions if you’re curious. I was wondering if there are any examples. I’m a non commutative person.
2026-03-27 13:48:35.1774619315
I was wondering what are some examples of G domains per kaplansky’s definition?
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As Kaplansky says (in Commutative Rings anyway) a Noetherian domain is a $G$-domain iff it is semilocal and has Krull dimension $1$. This gives lots of examples.
By starting with any Noetherian domain, you can take a minimal nonzero prime and localize at that, and the result is such a ring as described above. (Actually, I think you should be able to use any finite collection of minimal nonzero primes, and "semilocalize" at the intersection of their complements.)
Here is the DaRT search for $G$-domains, a.k.a. Goldman domains (I have excluded fields since those are trivial examples.) As you can see there are currently six positive hits, all of which fall into the category above, I believe.
A brief summary, for those who don't click through, there are things like this: $\mathbb Z_{(p)}$ and $\mathbb Z_S$ where $S=((2)\cup (3))^c$, $k[[x]]$ and $k[[x^2, x^3]]$.