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A well known quantum theorem revisited

The eigenvalue problem A mod i)= alpha i mod i) and the condition (A,B)=0 imply that (m mod B mod n)=0, under the restriction alpha m not= alpha n. However, it is observed that even in cases where (A,B) not=0 and the commutator obeys a certain functional form the simplest of which is (A,B)=c, then the off diagonal elements of B still vanish.