For a commutative ring with unity, I can find a nontrivial ring $R$ such that $R \times R \cong R$ as rings. For example you can just take $R = \prod_{n=1}^\infty A$ where $A$ is any nonzero ring. Is there an example of a ring where $R \times R \not\cong R$, but $R \times R \times R \cong R \times R$? Or is there a reason such a ring cannot exist? Here all the isomorphisms I'm considering are as rings.
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Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Feb 24 '23 at 19:51
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1Does this answer your question? Example of a ring such that $R^2\simeq R^3$, but $R\not\simeq R^2$ (as $R$-modules) – ℋolo Feb 24 '23 at 19:57
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As rings, or as $R$-modules? – Arturo Magidin Feb 24 '23 at 19:58
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I'm looking for as rings – determinecabinet Feb 24 '23 at 19:59
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Please put all the information on the post, not just in the comments. – Arturo Magidin Feb 24 '23 at 20:04
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Related: https://math.stackexchange.com/questions/1869512/does-a2-cong-b2-imply-a-cong-b-for-rings (In particular, any answer to this question gives an answer to that question with $A=R$ and $B=R^2$.) – Eric Wofsey Feb 24 '23 at 20:07