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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

Gregory Grant
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Javier Arias
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    This question is not substantially different from the one I've linked this question to. Please do not post questions that duplicate each other this closely. Thanks – rschwieb Sep 11 '15 at 15:57

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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.

rschwieb
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