A left ideal need not be a right $R$ module

Question

Let $R$ be a ring, it naturally follows that every left ideal of $R$ will also be a left $R$ module and an analogous result will also hold for right ideals of $R$.

I was trying to show that a left ideal may not be a right $R$ module and also vice versa. So for that i was wondering it would be suffienct to find examples of left ideals which are not right ideals and vice versa,but i can't seem to think of such examples, any ideas ?

Answer 1

What about

$$\left\{\;\begin{pmatrix}a&0\\b&0\end{pmatrix}\;\right\}\le M_2(\Bbb R)\;\;?$$

It's a left ideal of its ring, but not a right one.

Answer 2

Consider the free algebra $A=\mathbb{Z}\langle x,y\rangle$ generated by two elements $x,y$. The left ideal $Ax=\{ax\mid a\in A\}\subset A$ is obviously not a right ideal of $A$.