If $A$ is an ideal of $R$, show that $R/A$ is commutative if and only if $rs - sr \in A$ for all $r, s\in R$.

Question

How do I approach this question?

We know that $A$ is an ideal of $R$ and $r-s \in A$ where $r, s \in A$. Now, where do I go from here?

Answer 1

It follows immediately from the definitions: $$ rA \cdot sA = sA \cdot rA \iff rsA = srA \iff rs - sr \in A. $$

Answer 2

$R/A$ is commutative iff $xy = yx \,\,\,\forall x, y \in R/A$, i.e. iff $xy - yx = 0$ in $R/A$, i.e. iff $xy-yx \in A$.