Counterexample about scalar product of left ideals.

Question

Given two left ideals $I$ and $I'$ of a ring $R$ and any $r\in R$, then $Ir,I+I'$ are left ideals. I've proved both results and now I would like a counterexample for $rI$. I think that in general $rI$ is not a left ideal. The ring must be non-commutative.

Answer 1

Hint: a simple example of a noncommutative ring is $M_2(\mathbb{Z})$, the $2\times2$ matrices with integer entries. The matrices with zero second column form a left ideal $I$. Try now with various choices of $r$.