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Let $A$ be a ring, let $M_n(A)$ be the ring of $n$-by-$n$ matrices with elements in $A$, $A$ is Morita equivalent to $M_n(A)$, I was wondering if this also applied to infinite matrices? That is, if $A$ and $M_\infty(A)$ are Morita equivalent? Or if this is in general not true, are there cases where this happens? If at all.

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    $\begingroup$ What do you mean by infinite matrices? Some sort of finiteness is usually required (rows, columns, both...) $\endgroup$ Sep 27, 2013 at 5:28
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    $\begingroup$ For any one of the three options I mentioned, it is true that $M_\infty(M_\infty(A))$ is isomorphic to $M_\infty(A)$, so you get examples where you do have equivalence (isomorphism even!) On the other hand taking $A$ a field and column-finite matrices, for example, $M_\infty(A)$ has a non-trivial proper ideal (that of matrices with finitely many non-zero columns) while $A$ has none, so in that case you do not have a Morita equivalence. $\endgroup$ Sep 27, 2013 at 5:34
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    $\begingroup$ As I said, you need to impose some finiteness conditions (for example, that there be a finite number of non-zero coefficients in each row) for otherwise you simply do not get a ring! $\endgroup$ Sep 27, 2013 at 7:42
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    $\begingroup$ @Garlef: you mean the colimit, and these are along non-unital maps with the result being a non-unital ring. It's not entirely clear what the correct notion of module over such a thing is (especially in this case since the corresponding non-unital ring has an approximate identity which we might want to say something about). $\endgroup$ Sep 27, 2013 at 20:03
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    $\begingroup$ Sorry, but I give -1 since the question doesn't give the intended definition of $M_{\infty}(A)$, even after all the comments by Mariano (which get a +1). $\endgroup$ Sep 30, 2013 at 16:19

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No matter whether you mean row finite, column finite, or both the answer has to be no. For simplicity, let $R$ be a field. Then any of those rings has a proper 2-sided ideal consisting of the finite matrices (filled out with 0s), while the $M_n$ are all simple and a ring Morita equivalent to a simple ring is simple.

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