In the book that I read it says that A. Malcev gave such an example in 1936. But I can't find it... any help will be appriciated.
2026-04-04 00:15:00.1775261700
A ring without zero divisors that can't be extended to a ring with division
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Searching for "cannot be embedded in a division ring" with Malcev's name yielded this article, which cites the example on the final page.