From Milne's 'Fields and Galois Theory' chapter 01, if a commutative ring with unit contains a field as a subring, then we can consider asking what it's characteristic is. It has characteristic $p$ if it contains a prime field isomorphic to $\mathbb{F}_p$ and characteristic $0$ if it contains a prime field isomorphic to $\mathbb{Q}$.
Here a prime field is defined as a field isomorphic to $\mathbb{Q}$ or $\mathbb{F}_p$.
Then Milne claims that the prime field of any commutative ring $R$ with characteristic is unique. In other words, if such a ring has characteristic, then it is unique.
I understand how this works for the special case when $R$ is a field. Simply that the unique ring map from $f:\mathbb{Z}\to k$ has kernel in Spec$(\mathbb{Z})$ and hence principally generated by $0$ (if it is trivial) or some prime $p$ (if it is not trivial), after noting that $\mathbb{Z}/\ker(f)$ is subring of a field. Clearly, the kernel can't be both trivial and non-trivial, and further if it is not trivial then it is generated by the smallest (in terms of absolute values) non-zero element which is a unique prime. So, when $R$ is a field, then it has unique characteristic.
I have tried to expand this to the general case of $R$ being any commutative ring with characteristic. My initial thought was to consider the union of all fields it contains, but I found out that union of two subrings is a subring iff one contains the other. So, I wanted to consider the intersection of all fields $R$ contains, which is a subring that is closed under taking inverses, hence a field itself. In fact this is unique by construction. Is it possible to conclude from here? Any other proof strategies?