Characteristic of an integral domain: doubt in the proof Suppose by absurd that the characteristic of an integral domain is an integer $n$ not prime, say $n=n_1 \cdot n_2$.
Now we have $$na=(n_1 \cdot n_2)a=0 \implies (n_1 \cdot a) \cdot (n_2 \cdot a)=0 \implies n_1 \cdot a=0 \space \lor n_2 \cdot a=0$$
In my book it is considered a contradiction (for the minimality of $n$) but a priori it could be possible that $n_1$ and $n_2$ don't satisfy characteristic property.

Adopted definition of characteristic of an integral domain (with $charD \neq 0$): 
$$charD := min \{n \in \mathbb{N}| \space n \cdot a=0 \space \forall a \in D\}$$
In my book an integral domain is a commutative ring without non-zero zero divisors. I do not assume identity.

You still have proved that any vector will be killed by either one of $n_1$ and $n_2$. What you want is to be able to take the same $n_i$ for all vectors. 
So assume that there is a vector $a_1$ annihilated only by $n_1$ and a vector $a_2$ annihilated only by $n_2$.
Who kills $a_1+a_2$ then? If it is $n_1$ then $n_1(a_1+a_2)=0$ together with $n_1a_1=0$ proves that $n_1a_2=0$, which is a contradiction.
