# Is this exercise about field of quotients correctly stated?

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?

• Can you provide the authors definition of “domain”? – Knaus Aug 22 '20 at 9:09
• @Knaus A domain is a nontrivial ring without zero divisors. An integral domain is a commutative domain. – Akira Aug 22 '20 at 9:12
• 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. – Knaus Aug 22 '20 at 9:17
• @Knaus Your comment solved my problem. May you post it as an answer? – Akira Aug 22 '20 at 9:18
• 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". – Qi Zhu Aug 22 '20 at 10:24

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.