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  1. Does there exist a non-trival set $W$ such that $W\times W \cong W\times W\times W$ where $\times$ is the exterior product?

  2. Consider $R^2$ and $R^3$, I don't think they were isomorphism to each other, but I wasn't sure how to prove their order(i.e. suppose $|R|=\omega$)

  3. Could you find a subspace $W$ of $R$ or $C$(fields) such that $W$ was the smallest (in order) non trival set such that $W\times W \cong W\times W\times W$?

    • $\cong$ here meant for bijection, and was used to emphasize the another question: if $W$ was a field, could the bijection also be a ring homomorphism?
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    $\begingroup$ What do yuo mean by $\cong$, exactly? $\endgroup$ – Saucy O'Path Aug 15 '18 at 15:12
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    $\begingroup$ The product of two infinite numerable sets is numerable. $\endgroup$ – Tsemo Aristide Aug 15 '18 at 15:12
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    $\begingroup$ If $W$ is a field, then $W \times W$ and $W \times W \times W$ are vector spaces. There would be no vector space isomorphism because they have a different dimension over $W$, hence no ring isomorphism. $\endgroup$ – Joe Johnson 126 Aug 15 '18 at 15:17
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    $\begingroup$ Yes. If $X$ is an infinite set, then there is a set bijection between $X$ and $X^n$ for any $n \in \mathbb{N}$. $\endgroup$ – Joe Johnson 126 Aug 15 '18 at 15:22
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    $\begingroup$ @MaliceVidrine thank you. Existence of the bijection first, and then the isomorphism. Joe Johnson 126 has answered the question about the isomorphism already. $\endgroup$ – user416486 Aug 15 '18 at 15:25
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In the category of sets, any bijection $A \to A \times A$ is an isomorphism. In particular, take $A= \mathbb N$. Then define $f:\mathbb N \times \mathbb N \to \mathbb N$ by $(n,m) \mapsto 2^n \cdot 3^m$.


If $A$ is a field, there are some problems. For example, if you define $(a,b) \cdot (c,d)=(ac,bd)$, then $A \times A$ is most certainly not a field, since $(1,0) \cdot (0,1)=0$.


One way to "make this work" is that if you work in the category of $R$-modules, and consider $\otimes$ instead (which is reasonable, I think) then you can take $R \otimes_R R \cong R$ to be $(a \otimes b) \mapsto ab$ which is always an isomorphism of modules.

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  • $\begingroup$ Thank you. math.stackexchange.com/questions/2000739/… Patrick Stevens's answer provided a solution on $N$ by using Cantor pairing function, could it also be used on $R$? Further, what if $W$ is an non ordered field(A subspace in $C$)? $\endgroup$ – user416486 Aug 15 '18 at 15:50
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The problem of cancelletion of the direct product have been deeply studied and in most of cases finds an answer in the Krull-Remak-Schmidt Theorem.

A list of relevant links is the following:

Cancellation of Direct Product in Grp.

On cancellation in Groups.

Does Remak-Krull-Schmidt implies structure theorem for fg modules over PID.

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