A student is asked to prove that a quotient ring of a ring R modulo an ideal N is commutative if and only if (rs - sr) ∈ N for all r, s ∈ R. The student starts out: Assume R/ N is commutative. Then rs = sr for all r, s ∈ R/N.

a. Why does the instructor reading this expect nonsense from there on? 

b. What should the student have written? 

c. Prove the assertion. (Note the "if and only if.")