Let $R$ be an integral domain and consider the set $\mathbb{Z1}$ of all integral multiples of the identity element: $$\mathbb{Z1}=\{n1:n\in\mathbb{Z}\}.$$ It is required to prove that $\mathbb{Z1}$ is a field if and only if $R$ has positive characteristic.
The following is my attempt.
$(\leftarrow)$ Suppose $R$ has positive characteristic $p$. Then $p$ is prime as the characteristic of an integral domain is ether $0$ or a prime number. But $\mathbb{Z1}$ is a subdomain of $R$ since $R$ is an integral domain. Let $n1$ be a nonzero element in $\mathbb{Z1}$. Then $p1\in\mathbb{Z1}$ and since $\operatorname{gcd}(n,p)=1$ we have $1=na+pb=(na+pb)1=n(a1)+b(p1)=n(a1)=(n1)(a1)$ for some $a,b\in\mathbb{Z}$. Thus $n1$ is invertible and hence $\mathbb{Z1}$ is a field.
When proving the forward direction I was stuck. The idea I had was that if $\mathbb{Z1}$ is a field (and thus a subdomain of $R$) then $\operatorname{Char} \mathbb{Z1}=\operatorname{Char}R$ due to the fact that characteristic of an integral domain is either $0$ or a prime number. I cannot find a way to proceed. Could someone please help? Thanks.
