Characteristic of a product ring

Detailed Solution

The characteristic of $R$ is the smallest positive integer $k$ such that $k\cdot(1,1)=(0,0)$, i.e. $k\cdot1=0$ in both $\mathbb Z_4$ and $\mathbb Z_6$ simultaneously. This requires $k$ to be a multiple of $\operatorname{char}(\mathbb Z_4)=4$ and of $\operatorname{char}(\mathbb Z_6)=6$, so the smallest such $k$ is $\operatorname{lcm}(4,6)=12$. Thus the correct option is $\boxed{\mathrm{B}}$.