Morita equivalence between right and left ideals of a ring

Question

I would like to know whether Morita equivalence is a useful tool when dealing with right and left ideals of a ring. If so, could someone illustrate it on the example of $2\times 2$ matrices?

Thanks

Answer 1

Morita equivalence is not very useful when dealing with right and left ideals.

For example, $\Bbb R$ and $M_2(\Bbb R)$ are Morita equivalent, and the first one has exactly two left ideals, while the second has infinitely many left ideals. You can see this destroys any hopes for correspondence of one-sided ideals.