(January 2009) Let \(I\subseteq R\) be an ideal of \(R\), and suppose that \(xy-yx\in I\) for every \(x,y\in R\). Prove that the quotient ring \(R/I\) is commutative.
Solution
Proof. GivenI +a,I +b∈R/I, our assumption on I implies that ab−ba∈I. By the coset relation, we obtain I +ab =I +ba. Then (I +a)(I +b) =I +ab =I +ba = (I +b)(I +a), so R/I is commutative, as desired QED