The number 2 in a PID

Question

Let $R$ be a PID. Then $R$ is a commutative ring with multiplicative identity $1$. We can then define $2=1+1$. From here, what is known about $2$ and its prime factors? I suppose this breaks into two questions. If $2=0$, then what can I conclude, if anything, about the structure of $R$? For example, finite fields $\mathbb{F}_{2^k}$ are PIDs with characteristic $2$, but are there possibly other PIDs with characteristic $2$? If I know that $2\neq 0$, then what can I tell about the divisors of $2$? In quadratic integer rings that are PIDs such as $\mathbb{Z}[i]$ and $\mathbb{Z}[\sqrt{-2}]$, we know that $2$ must be prime, a product of two nonassociated primes, or associated to the square of a prime. Is there an analogous fact in the general case? Any information would be greatly appreciated.

Answer 1

If $F$ is a field of characteristic $2$, then $F[X]$ is a PID of characteristic $2$.

When $2\neq 0$, there may be no prime divisors of $2$, because it may be a unit. This happens in any domain that has characteristic greater than $2$.

Answers to your last question will probably be too complicated in full generality. It is probably a large part of what algebraic number theory is about.