There is a classical theorem that [no field can be expressed as finite union of proper subfields](https://mathoverflow.net/questions/39605/).


In contrast, there is an [example](http://matwbn.icm.edu.pl/ksiazki/cm/cm7/cm717.pdf) of an integral domain that can be expressed as finite union of proper subrings. 


Therefore, I wonder whether there are any known results about the existence of Principal ideal domain (or even Euclidean domain) that can be expressed as finite union of proper subrings?



PID expressed as finite union of subrings
Consider the following subrings of $\mathbb{F}\_2^3$:
$$
V\_1=\{(0,0,0),(1,1,1),(1,0,0),(0,1,1)\}\\
V\_2=\{(0,0,0),(1,1,1),(0,1,0),(1,0,1)\}\\
V\_3=\{(0,0,0),(1,1,1),(0,0,1),(1,1,0)\}
$$
Then $\mathbb{F}\_2^3=V\_1\cup V\_1\cup V\_3$.


Let $R$ be a PID admitting $3$ distinct surjective ring maps $f\_1,f\_2,f\_3$ onto $\mathbb{F}\_2$
and write
$$
R\_i=\{a\in R: (f\_1(a),f\_2(a),f\_3(a))\in V\_i\}
$$
for $i=1,2,3$.
Then $R\_1,R\_2,R\_3$ are subrings of $R$ with $R=R\_1\cup R\_2\cup R\_3$. They are proper subrings by the Chinese Remainder Theorem.


To construct such $R$, take a cubic extension $K$ of $\mathbb{Q}$ in which $2$ splits into a product of $3$ primes $P\_1,P\_2,P\_3$, and choose $R$ to be the localization of $\mathcal{O}\_K$ at the multiplicative set $R\setminus(P\_1\cup P\_2\cup P\_3)$.


