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I'm doing Exercise 6 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.

Show that any monomorphism $D \rightarrow D^{\prime}$ of domains yields a corresponding monomorphism $\operatorname{Quot}(D) \rightarrow \operatorname{Quot}\left(D^{\prime}\right)$ of fields.

In this textbook and on this wikipedia page, the field of quotients is constructed from integral domain, not just domain. As such, I suspect that the authors mean integral domain rather than domain.

Could you please verify if my understanding is fine?

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  • $\begingroup$ Can you provide the authors definition of “domain”? $\endgroup$ – Knaus Aug 22 '20 at 9:09
  • $\begingroup$ @Knaus A domain is a nontrivial ring without zero divisors. An integral domain is a commutative domain. $\endgroup$ – Akira Aug 22 '20 at 9:12
  • $\begingroup$ I imagine there is a mistake then. There are constructions generalizing field of fractions, but it obviously won’t be a field in the non commutative case, (you want there to be an inclusion of a non commutative ring) so it should rather be called the skew field of fractions. $\endgroup$ – Knaus Aug 22 '20 at 9:17
  • $\begingroup$ @Knaus Your comment solved my problem. May you post it as an answer? $\endgroup$ – Akira Aug 22 '20 at 9:18
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    $\begingroup$ I have checked the book, the authors never defined just "domain" in the way you did. They use the word "domain" with the same meaning as "integral domain". $\endgroup$ – Qi Zhu Aug 22 '20 at 10:24
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If their definition of Quot is such that for a domain (not necessarily commutative) $R$ there is a monomorphism $R\rightarrow \mathrm{Quot}(R)$, then clearly Quot$(R)$ is not necessarily a field, as it is non-commutative whenever $R$ is.

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    $\begingroup$ For future readers, let me note that this particular textbook uses the notions "domain" and "integral domain" synonymously. $\endgroup$ – Qi Zhu Aug 22 '20 at 10:27

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