I have been thinking a bit about Morita equivalence http://en.wikipedia.org/wiki/Morita_equivalence and I would like to know whether it also applies to subrings such as right or left ideals. And, if so, how specifically? Could you illustrate if with the example of 2x2 matrices with zero entries in the bottom row? Since Morita equivalence concerns non-commutative rings, I am assuming it should apply, but I am just guessing.
Thanks in advance.
A right ideal of a ring $R$ is the same thing as a submodule of $R$ as a right $R$-module, which is in turn the same thing as a(n equivalence class of) monomorphism(s) $M \to R$ in the category of right $R$-modules. Since this is a categorical concept, it is preserved under equivalences of categories, and hence under Morita equivalences. For example, using the Morita equivalence between $R$ and $M_n(R)$ you conclude that right ideals of $M_n(R)$ are the same thing as submodules of $R^n$ as a right $R$-module.