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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.

Javier Arias
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  • I do not know if this might be useful for the thread: https://www.math.washington.edu/~smith/Research/morita-weyl.pdf http://www.math.ohiou.edu/~jain/044.pdf – Javier Arias May 28 '15 at 11:11
  • I do not understand what you are asking: what do you mean by «Morita theory applies to subrings»? – Mariano Suárez-Álvarez May 28 '15 at 18:13
  • Quite simple: Morita equivalence refers to Rings. Ideals are subrings with respect to a ring. I just want to know whether that same Morita equivalence applies in that case. According to the answer below, it seems it does. – Javier Arias May 28 '15 at 18:16
  • I still don't understand. In fact, the explanation in your comment is less comprehensible to me that the original question. I don't know what you mean by Morita equivalence applying «to that case». – Mariano Suárez-Álvarez May 29 '15 at 02:30

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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.

Qiaochu Yuan
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  • Could you illustrate that for the case of 2x2 matrices with zero entries in the bottom row? Thanks – Javier Arias May 28 '15 at 18:14
  • @Javier: what Morita equivalence do you have in mind in that case? – Qiaochu Yuan May 28 '15 at 18:15
  • That of the 2x2 matrices with zero entries in the bottom row with regard to, for instance, all 2x2 matrices, or all quadratic matrices, or even N or Z – Javier Arias May 28 '15 at 18:39
  • @Javier: I don't understand what you're asking. What is it that you want illustrated? – Qiaochu Yuan May 28 '15 at 18:40
  • a 2x2 matrix with zeroes in the bottom row is an ideal of the whole set of 2x2 matrices or of the whole set of quadratic matrices....and that ideal is a subring.....so I want to how if and if so, in which way, Morita equivalence applies to that case with subrings instead of rings – Javier Arias May 28 '15 at 19:04
  • @Javier: Morita equivalence has nothing to say about subrings (and here you mean non-unital subrings). Subrings are not invariant under Morita equivalences. – Qiaochu Yuan May 28 '15 at 19:30
  • But above you mentiones that 2x2 matrices are Morita invariant....or did I get something wrong – Javier Arias May 28 '15 at 19:39
  • @Javier: it's true that $R$ and $M_2(R)$ are Morita equivalent, and as I say in my answer, you can use this to deduce things about left or right (even two-sided) ideals in $M_2(R)$ from corresponding facts about $R$. That's because these can all be described categorically in categories of modules. But subrings don't have this property. – Qiaochu Yuan May 28 '15 at 20:17
  • But M2*R( is a subring, right....if so, it has that property then... – Javier Arias May 28 '15 at 20:30
  • «you conclude that right ideals of $M_n(R)$ are the same thing as submodules of $R^n$ as a right $R$-module» I don t understand this. "The same thing" means something unorthodox here :-) – Mariano Suárez-Álvarez May 29 '15 at 02:29
  • @Mariano: "the same thing" means the usual thing here. I mean that the Morita equivalence induces an isomorphism of posets (say) between right ideals of $M_n(R)$ and right submodules of $R^n$. – Qiaochu Yuan May 29 '15 at 17:48