TITLE: PID expressed as finite union of subrings
QUESTION [12 upvotes]: There is a classical theorem that no field can be expressed as finite union of proper subfields.
In contrast, there is an example 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?

REPLY [10 votes]: 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 $R\setminus(P_1\cup P_2\cup P_3)$.