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What are some good examples of algebraic theorems that have no known algebraic proofs? A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...

Preface: I know very little about noncommutative algebra and noncommutative geometry, so please feel free to make improvement suggestions for my question. Also, to my knowledge there are several ...

I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...