Is there an associative unital algebra $A$ which is isomorphic to its own algebra of $n^2\times n^2$ matrices $\operatorname{Mat}_{n^2}(A)$, but not isomorphic to its algebra of $n \times n$ matrices $\operatorname{Mat}_n(A)$?
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asked Nov 4 at 23:21
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1$\begingroup$ Leavitt for any pair of integers m<n a ring where the free module on m generators is isomorphic to the free module on n generators and this is the first isomorphism to appear. So if you take 1, n^2 that should likely do it. $\endgroup$– Benjamin SteinbergCommented Nov 4 at 23:42
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1$\begingroup$ Actually maybe this example doesn’t quite work by the result of degruyter.com/document/doi/10.1515/CRELLE.2008.082/html $\endgroup$– Benjamin SteinbergCommented Nov 4 at 23:50
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