There is a conclusion like this:
Let $R$ be a commutative ring. Let $n=|R|$ be squarefree ($\nu_p(n)<=1$ for all primes $p$). Then its characteristic is $n$.
Now, let's say I have a commutative ring $R$ with size $2^2\cdot 3^3 \cdot 5^4$. Does its characteristic $d$ must contain as factor at least each of $\{2,3,5\}$?
I believe the logic why ring characteristic with squarefree size $n$ must contain each of $n$'s prime factor at least once still hold to ensure the minimality of characteristic.
Am I right?