If R has a non zero commutative left ideal then R is commutative

Question

Let $R$ be a prime ring. If $R$ contains a commutative nonzero left ideal, then $R$ is commutative.

my approach is that

Let $U$ be a nonzero commutative left ideal. Let $x$ in $U$ and $l$, $m$ in $R$. then $lx$ in $U$ and $mx$ in $U$. As given $U$ is commutative. $lx. mx$=$mx.lx$. Help me to show that $lm$=$ml$

Answer 1

Not true. Consider e.g. the ring of $3 \times 3$ matrices of the form $\pmatrix{a & 0 & 0\cr 0 & b & c\cr 0 & d & e\cr}$, with two-sided commutative ideal consisting of elements of the form $\pmatrix{a & 0 & 0\cr 0 & 0 & 0\cr 0 & 0 & 0\cr}$.