# Can $[A\otimes I_m,B]=0$ for all $A$ imply $B=I_n\otimes C$?

Suppose that $$A$$ is an $$n\times n$$ matrix and $$I_m$$ is an $$m-$$dimensional identity matrix. If $$[A\otimes I_m,B]=0$$ for all $$A$$ where $$\otimes$$ is Kronecker product, does it imply $$B$$ must be of form $$B=I_n\otimes C$$ where $$C$$ is an $$m\times m$$ matrix? If so, how to prove it?

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