Spin 1/2 particles hamiltonian, addition of angular momentum confusion

Suppose I want to compute $S^{1}_z -S^{2}_z$ on a singlet state $|0,0>$. (where $S^{i}_z$ are two particles' spin operators). $$|0,0> = \frac{1}{\sqrt{2}} (|\frac{1}{2},-\frac{1}{2}> - |-\frac{1}{2},\frac{1}{2}>).$$ I don't get the correct answer by just computing and using $$S^{1}_z|m_{s1}, m_{s2}> = m_{s1}|m_{s1}, m_{s2}>.$$ I know how to get the answer by matrix representation but how can I use this operators?

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