Description: Enumerate the primes in $\mathbb N$. Let $M$ be the additive submonoid of positive rationals generated by $\frac{1}{2^ip_i}$ $i \geq 0$. With field $F$ and indeterminate $X$, and generate $F$ algebra generated by $X^m$, $m\in M$. Localize at the set of elements with nonzero constant term. This localization is the ring.

Notes:

Keywords semigroup ring

Reference(s):

- Anne Grams, Atomic Rings And The Ascending Chain Condition For Principal Ideals., Math. Proc. Of The Cambridge Phil. Soc. Vol. 75. No. 03. Cambridge University., (1974). Main Example

The ring lacks the following properties:

algebraically closed field
Artinian
characteristic 0 field
Dedekind domain
dual
Euclidean domain
Euclidean field
field
finite
Frobenius
Kasch
Noetherian
ordered field
perfect
perfect field
principal ideal domain
principal ideal ring
pseudo Frobenius
Pythagorean field
quadratically closed field
self-injective
semiprimary
semisimple
von Neumann regular

We don't know if the ring has or lacks the following properties:

ACC principal
atomic domain
Bezout domain
Bezout ring
clean
Cohen-Macaulay
coherent
continuous
distributive
finitely pseudo Frobenius
GCD domain
Gorenstein
Jacobson (Hilbert)
Krull domain
local
local complete intersection ring
Mori domain
normal
normal domain
Prufer domain
rad-nil
regular
regular local
Schreier domain
semilocal
semiperfect
semiprimitive
semiregular
serial
stable range 1
strongly pi regular
unique factorization domain
valuation