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?
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 $|R|=\omega$)
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?
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.
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: